Model Comparison: Functions

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Gemini 3 Pro

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What You Need to Know

• Notation is key: Understanding f(x) notation is fundamental; remember that x is the input and the entire expression f(x) represents the output (y-value).
• Graphs tell the story: You must be able to link an equation to the shape of its graph (e.g., y=x^2 is a parabola, y=2^x is a curve approaching an asymptote) without a graphing calculator.
• Transformations are predictable: Shifts, stretches, and reflections follow strict rules that apply to all function types, not just one.
• Domain restrictions: Always check for division by zero and negative numbers under even roots.

Function Definition and Notation

The Core Concept

A function is a relation where every input (x) produces exactly one output (y). Visually, a graph represents a function if it passes the Vertical Line Test: no vertical line can intersect the graph at more than one point.

Function Notation

Standard notation is f(x), read as "f of x."

• x = Input (Independent Variable)
• f(x) = Output (Dependent Variable, usually y)

Example:
If f(x) = 3x^2 - 4, find f(-2).
f(-2) = 3(-2)^2 - 4
f(-2) = 3(4) - 4
f(-2) = 8

Domain and Range

• Domain: The set of all possible inputs (x-values) for which the function is defined. Watch out for denominators (cannot be zero) and even roots (radicand must be non-negative).
• Range: The set of all resulting outputs (y-values).

Composite Functions

Composition involves plugging one function into another. Notation includes f(g(x)) or (f \circ g)(x).

Example:
If f(x) = 2x + 1 and g(x) = x^2, find f(g(3)).

1. Find g(3): 3^2 = 9
2. Plug result into f: f(9) = 2(9) + 1 = 19

Exam Focus

• Why it matters: Functions form the backbone of the ACT Math section (Intermediate Algebra). You will see 4-6 questions specifically testing notation and evaluation.
• Typical question patterns:
  • "Given f(x) = …, what is f(a+b)?"
  • "For which value of x is f(x) undefined?" (Check denominators).
  • Nested functions: g(h(f(2))).
• Common mistakes:
  • Confusing f(2) (plugging in 2) with 2f(x) (multiplying the result by 2).
  • Assuming the domain is always "all real numbers" without checking for restrictions.

Linear Functions

Slope-Intercept Form

The most common form on the ACT is f(x) = mx + b.

• m = Slope (Rate of change, \frac{\text{rise}}{\text{run}}, \frac{y2 - y1}{x2 - x1})
• b = y-intercept (Value of f(x) when x=0)

Parallel and Perpendicular Lines

• Parallel Lines: Have the same slope (m1 = m2).
• Perpendicular Lines: Have negative reciprocal slopes (m1 = -\frac{1}{m2}). Their product is -1.

Exam Focus

• Why it matters: Linear functions are the easiest points to gain but also easy to lose if you rush algebraic manipulation.
• Typical question patterns:
  • Word problems interpreting slope as a "rate" (e.g., cost per mile).
  • Finding the equation of a line given two points.
• Common mistakes:
  • Flipping x and y in the slope formula.
  • Forgetting that a vertical line (x = k) has an undefined slope and is not a function.

Polynomial Functions

Quadratic Functions

A polynomial of degree 2, typically written as f(x) = ax^2 + bx + c. The graph is a parabola.

• Vertex: The peak or valley of the parabola. The x-coordinate is at x = -\frac{b}{2a}. Plug this x back in to find y.
• Roots/Zeros: The x-intercepts where f(x) = 0. Solve by factoring or using the Quadratic Formula:
  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Higher-Degree Polynomials

• Degree: The highest exponent determines the shape.
  • Odd degree (like x^3): Ends go in opposite directions.
  • Even degree (like x^4): Ends go in the same direction.
• Leading Coefficient:
  • Positive: Right side goes up.
  • Negative: Right side goes down.

Exam Focus

• Why it matters: Quadratics appear frequently. You must be comfortable factoring and finding the vertex quickly.
• Typical question patterns:
  • "What are the solutions to…" (Find roots).
  • "What is the maximum height…" (Find the vertex y-coordinate).
• Common mistakes:
  • Forgetting the 2a in the denominator of the quadratic formula.
  • Sign errors when calculating the discriminant b^2 - 4ac.

Radical Functions

Basics

The parent function is f(x) = \sqrt{x}.

• Graph shape: Half a parabola on its side, starting from (0,0) and extending to the right.
• Domain: For square roots (and other even roots), the argument must be \ge 0.

Example:
Find the domain of f(x) = \sqrt{2x - 6}.
Set 2x - 6 \ge 0 \rightarrow 2x \ge 6 \rightarrow x \ge 3.

Exam Focus

• Why it matters: These questions usually test domain restrictions.
• Typical question patterns:
  • "For which values of x is the function real-valued?"
• Common mistakes:
  • Thinking square roots can output negative numbers (the principal root function \sqrt{x} is always non-negative).

Piecewise Functions

Definition

A function defined by different formulas for different intervals of the domain.

Example:
f(x) = \begin{cases} 2x & \text{if } x < 0 \ x^2 & \text{if } x \ge 0 \end{cases}

To evaluate f(-3), check the condition: -3 < 0, so use the top rule: 2(-3) = -6.
To evaluate f(5), check the condition: 5 \ge 0, so use the bottom rule: 5^2 = 25.

Exam Focus

• Why it matters: Students often panic when they see the bracket notation, but the math is simple substitution.
• Typical question patterns:
  • "Given the piecewise function below, what is f(4)?"
  • Graphs with open circles (< or >) vs. closed dots (\le or \ge) at the boundaries.
• Common mistakes:
  • Plugging the x-value into both equations instead of selecting the correct one based on the inequality.

Exponential and Logarithmic Functions

Exponential Functions

Form: f(x) = a \cdot b^x where b > 0 and b \neq 1.

• Growth: If b > 1, the graph rises rapidly to the right.
• Decay: If 0 < b < 1, the graph falls to the right, approaching zero.
• Asymptote: The graph approaches but never touches the horizontal line (usually y=0 unless shifted).

Logarithmic Functions

Logarithms are the inverses of exponentials: y = \log_b(x) is equivalent to b^y = x.

• Domain: You can only take the log of a positive number (x > 0).
• Key Properties:
  • Product Rule: \logb(xy) = \logb(x) + \log_b(y)
  • Quotient Rule: \logb(\frac{x}{y}) = \logb(x) - \log_b(y)
  • Power Rule: \logb(x^n) = n \cdot \logb(x)

Exam Focus

• Why it matters: The ACT loves testing the conversion between log form and exponent form.
• Typical question patterns:
  • "If 5^x = 12, write x in terms of logs." (x = \log_5(12)).
  • Simplifying log expressions using properties.
  • Compound interest problems: A = P(1 + r)^t.
• Common mistakes:
  • Thinking \log(a+b) = \log a + \log b (This is FALSE).
  • Forgetting that \log x (with no base) implies base 10, and \ln x implies base e.

Function Transformations and Translations

The Golden Rules of Shifting

If you know what y=f(x) looks like, you can graph variations by identifying h and k in y = a \cdot f(x - h) + k.

Note: Horizontal shifts are counter-intuitive. x - 3 moves Right, x + 3 moves Left.

Exam Focus

• Why it matters: This allows you to solve graphing problems without actually plotting points.
• Typical question patterns:
  • "The graph of y=x^2 is shifted 3 units right and 2 units down. What is the new equation?" (Answer: y = (x-3)^2 - 2).
• Common mistakes:
  • Moving left for f(x-2) instead of right.

Analyzing Graphs and Key Features

Intercepts

• y-intercept: Set x=0 and solve for y. Every function (that is defined at 0) has exactly one.
• x-intercepts (Zeros): Set y=0 and solve for x. A function can have none, one, or many.

Intervals

• Increasing: As x gets bigger (move right), y goes up.
• Decreasing: As x gets bigger (move right), y goes down.

Asymptotes

Lines that a graph approaches but doesn't touch.

• Vertical Asymptotes: Occur where the denominator is zero (rational functions).
• Horizontal Asymptotes: Occur in exponential functions or rational functions based on end behavior.

Exam Focus

• Why it matters: ACT questions often show a graph and ask you to identify the equation, or vice-versa.
• Typical question patterns:
  • Matching an equation to a graph based on roots and intercepts.
  • Identifying the range of a function from its picture.
• Common mistakes:
  • Confusing the x-coordinate of the vertex with the axis of symmetry.

Quick Review Checklist

• Can you evaluate a composite function like f(g(2)) from the inside out?
• Do you know the difference between a domain (inputs) and a range (outputs)?
• Can you identify the slope and y-intercept instantly from y = mx + b?
• Do you know how to find the vertex of a parabola given in standard form y = ax^2 + bx + c?
• Can you rewrite a logarithmic equation y = \log_b(x) as an exponential equation b^y = x?
• Do you know that f(x-5) shifts a graph to the right, not the left?
• Can you determine the domain of a radical function like \sqrt{x-4}?
• Do you know that parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes?

Final Exam Pitfalls

1. The "Undefined" Trap:

   • Mistake: Assuming a function exists everywhere.
   • Correction: Always check for division by zero (\frac{1}{0}) or square roots of negative numbers. These values are excluded from the domain.

2. The Parentheses Error:

   • Mistake: Typing -3^2 into a calculator instead of (-3)^2.
   • Correction: -3^2 is -9; (-3)^2 is 9. Function inputs often act like they are in parentheses.

3. The Horizontal Shift Switch:

   • Mistake: Seeing y = (x+4)^2 and thinking the graph moves 4 units to the positive side.
   • Correction: Remember "inside is opposite." x+4 moves the graph to the negative side (Left 4).

4. Slope Confusion:

   • Mistake: Calculating slope as \frac{x2 - x1}{y2 - y1}.
   • Correction: It is always Rise over Run (\frac{y}{x}). Think "y goes to the sky."

5. f(x) is just y:

   • Mistake: Getting confused by the notation f(x) and trying to solve for f.
   • Correction: Mentally replace f(x) with y whenever you see it. It is just a label for the output variable.

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GPT 5.2 Pro

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What You Need to Know

• A function assigns each input in its domain to exactly one output—ACT questions often test domain/range, evaluating f(x), and interpreting graphs.
• Most function problems reduce to spotting key features (intercepts, vertex/turning points, end behavior, asymptotes) and using them to match equations ↔ graphs.
• Transformations (shifts, stretches, reflections) let you rewrite and sketch graphs quickly—frequently tested via “how does the graph change?” questions.
• Expect functions to appear throughout ACT Math (especially the ACT’s “Preparing for Higher Math” skills), often embedded inside algebra, modeling, and graph interpretation.

Function Definition and Notation

A function is a relation where every input has exactly one output.

Key terms

• Domain: all allowable inputs x.
• Range: all possible outputs y.
• Function notation: f(x) means the output when the input is x.

Notation reference

Evaluating and solving with functions

• Evaluate: if f(x)=2x-3, then f(5)=2(5)-3=7.
• Solve: if f(x)=7, solve 2x-3=7.

Domain basics (common ACT patterns)

• Fractions: denominator cannot be 0.
• Radicals (even root): radicand must be \ge 0.
• Logs: argument must be >0.

Worked example

Given f(x)=\dfrac{x+1}{x-2}, find the domain.
1) Denominator restriction: x-2 \ne 0
2) So x \ne 2
Domain: all real x except 2.

Exam Focus

• Why it matters: Function notation and domain/range are core ACT “language” used across algebra and graph questions.
• Typical question patterns:
  • “If f(x)=\dots, find f(\text{number})” or “solve f(x)=\text{number}.”
  • Identify whether a relation is a function from a table/graph.
  • Find domain restrictions from a formula.
• Common mistakes:
  • Treating f(x) as f\cdot x—it’s a single symbol.
  • Forgetting domain restrictions (especially denominators and logs).
  • Assuming a graph is a function without checking the “one output per input” idea.

Linear Functions

A linear function has constant rate of change (slope) and graphs as a line.

Forms you should recognize

• Slope-intercept: y=mx+b (slope m, y-intercept b)
• Point-slope: y-y1=m(x-x1)

Slope and intercepts

• Slope from two points (x1,y1) and (x2,y2) :
  m=\dfrac{y2-y1}{x2-x1}
• x-intercept: set y=0.
• y-intercept: set x=0.

Worked example

Line through (2,5) with slope -3.
1) Use point-slope: y-5=-3(x-2)
2) Simplify: y-5=-3x+6
3) y=-3x+11

Real-world link

Linear models fit constant-rate situations: hourly pay, constant-speed travel, steady growth/decay per unit.

Exam Focus

• Why it matters: Linear functions are among the most frequent function types on ACT due to quick algebra/graph interpretation.
• Typical question patterns:
  • Find slope/intercepts from an equation or two points.
  • Match a line’s equation to its graph (sign of slope, intercept size).
  • Interpret slope as a rate (units).
• Common mistakes:
  • Mixing up x- vs y-intercepts (remember: intercept is where the other variable is 0).
  • Sign errors when computing slope.
  • Forgetting to distribute in y-y1=m(x-x1).

Polynomial Functions

A polynomial function is f(x)=anx^n+a{n-1}x^{n-1}+\dots+a1x+a0 with nonnegative integer exponents.

Key graph features

• Degree: highest exponent n.
• Leading coefficient a_n controls end behavior.
• Zeros/roots: where f(x)=0 (the x-intercepts).
• Multiplicity idea (ACT-level): repeated factors may make the graph “touch” and turn instead of crossing.

Factored form helps

If f(x)=(x-1)(x+2), zeros are x=1 and x=-2.

Worked example

Find zeros of f(x)=x^2-5x+6.
1) Factor: x^2-5x+6=(x-2)(x-3)
2) Set equal to 0: (x-2)(x-3)=0
3) Solutions: x=2 or x=3

Exam Focus

• Why it matters: Polynomials appear in equation solving, graph matching, and modeling (especially quadratics).
• Typical question patterns:
  • Factor to find zeros/intercepts.
  • Use degree/leading coefficient to predict end behavior.
  • Compare graphs of polynomials with different degrees.
• Common mistakes:
  • Sign errors when factoring (e.g., constant term).
  • Confusing zeros with y-intercept (compute f(0) for that).
  • Ignoring end behavior clues when choosing a matching graph.

Radical Functions

A radical function involves a root, commonly f(x)=\sqrt{x} or f(x)=\sqrt{ax+b}.

Domain and shape

• For even roots: require \text{radicand} \ge 0.
• Graph of y=\sqrt{x} starts at (0,0) and increases, concave down.

Worked example

Find the domain of f(x)=\sqrt{2x-8}.
1) Require radicand \ge 0: 2x-8 \ge 0
2) Solve: 2x \ge 8 so x \ge 4
Domain: x \ge 4.

Real-world link

Square-root models show up in geometry (side lengths from area), physics (some distance–time relationships), and statistics (standard deviation formulas).

Exam Focus

• Why it matters: ACT often tests domain restrictions and graph shifts of radicals.
• Typical question patterns:
  • Find domain or evaluate at a given x.
  • Solve equations with radicals (often by squaring both sides).
  • Identify a shifted/scaled square-root graph.
• Common mistakes:
  • Forgetting the radicand constraint for domain.
  • Introducing extraneous solutions after squaring—always check in the original equation.
  • Misapplying transformations (inside vs outside the radical).

Piecewise Functions

A piecewise function defines different rules on different intervals of the domain.

How to evaluate

1) Locate which condition your input satisfies.
2) Use that rule only.

Worked example

Let
f(x)=\begin{cases}2x+1 & x

• x=-3<0: f(-3)=2(-3)+1=-5
• x=2 is in 0\le x\le 2: f(2)=2^2=4
• x=5>2: f(5)=3

Real-world link

Tax brackets, shipping costs, and utility billing often use piecewise rules.

Exam Focus

• Why it matters: ACT uses piecewise functions to test careful reading and graph interpretation.
• Typical question patterns:
  • Evaluate f(a) from a piecewise definition.
  • Identify jump/closed/open circles on graphs at boundary points.
  • Solve f(x)=k by checking each interval.
• Common mistakes:
  • Using the wrong interval rule.
  • Mishandling endpoints (e.g., confusing < vs \le).
  • Forgetting that boundaries may have different left/right behavior.

Exponential and Logarithmic Functions

An exponential function has the variable in the exponent, typically f(x)=a\cdot b^x with b>0 and b\ne 1.
A logarithmic function is the inverse type of exponential and is defined where its argument is positive.

High-value facts

• Growth if b>1, decay if 0<b<1.
• Exponential graphs often have a horizontal asymptote (commonly y=0 if no vertical shift).
• Log domain: argument must satisfy >0.

Worked example (exponential)

If f(x)=3\cdot 2^x, find f(4).
1) Substitute: f(4)=3\cdot 2^4
2) Compute: 2^4=16
3) f(4)=48

Real-world link

Population growth, compound interest, depreciation, and half-life are exponential models.

Exam Focus

• Why it matters: These functions appear in modeling and graph-feature questions (growth/decay, asymptotes, domain).
• Typical question patterns:
  • Compare growth vs decay from the base b.
  • Evaluate exponential expressions quickly.
  • Determine allowed inputs for logs (argument >0) or identify asymptotes/shifts.
• Common mistakes:
  • Treating b^x like x^b—they behave very differently.
  • Ignoring the log domain restriction.
  • Missing the effect of vertical shifts on asymptotes.

Function Transformations and Translations

A transformation changes a base graph’s position or shape.

Common transformations (start from y=f(x))

Memory aid

• Inside changes (to x) move the graph opposite the sign: f(x-h) shifts right by h.
• Outside changes (added after) move the graph in the same direction: f(x)+k shifts up by k.

Worked example

Describe transformations from y=\sqrt{x} to y=\sqrt{x-4}+3.

• x-4 inside: shift right 4.
• +3 outside: shift up 3.

Exam Focus

• Why it matters: ACT often asks you to match an equation to a transformed parent graph without heavy computation.
• Typical question patterns:
  • Identify shifts/reflections from a formula.
  • Given a graph of f, sketch/identify graph of f(x)+k or f(x-h).
  • Determine new intercepts or starting points after a shift.
• Common mistakes:
  • Reversing horizontal shift direction.
  • Forgetting reflections when a negative sign appears.
  • Applying the shift to the wrong feature (e.g., moving the asymptote incorrectly).

Analyzing Graphs and Key Features of Functions

ACT graph questions reward quick identification of structure rather than perfect sketching.

Features to read off or compute

• Intercepts: solve f(x)=0 for x-intercepts; compute f(0) for y-intercept.
• Increasing/decreasing intervals: where outputs rise/fall as x increases.
• Maximum/minimum (often quadratics): vertex gives extremum.
• Asymptotes (common in rational/exponential/log): lines the graph approaches.
• End behavior: what happens as x\to \infty or x\to -\infty (often from degree and leading coefficient for polynomials).

Fast matching tips (equation ↔ graph)

• For lines, use slope sign and intercept.
• For quadratics, look for vertex location and whether it opens up/down.
• For exponentials, identify growth/decay and horizontal asymptote.
• For radicals, look for an “endpoint” start (often where the radicand becomes 0).

Worked example

A quadratic has vertex (1,-4) and passes through (0,-3) . Find an equation.
1) Vertex form: y=a(x-1)^2-4
2) Use point (0,-3) : -3=a(0-1)^2-4
3) -3=a-4 so a=1
4) Equation: y=(x-1)^2-4

Exam Focus

• Why it matters: Many ACT items are “graph literacy” questions—quickly extracting features and choosing the matching option.
• Typical question patterns:
  • Identify intercepts, vertex, or asymptote from a graph.
  • Choose which equation could generate a given graph.
  • Interpret a real-world graph (rate, increases/decreases, max/min).
• Common mistakes:
  • Reading scales incorrectly on axes.
  • Confusing an intercept with a point near the axis.
  • Missing asymptotes and assuming the function “reaches” the line.

Quick Review Checklist

• Can you explain what makes a relation a function and how to check it from a graph or table?
• Can you evaluate f(a) correctly by substitution and simplify without algebra errors?
• Do you know how to find domain restrictions for fractions, radicals, and logarithms?
• Can you find slope and write a linear equation from a point and a slope (or two points)?
• Can you factor a polynomial to find zeros and interpret them as x-intercepts?
• Can you determine the domain of a radical function by solving an inequality?
• Can you evaluate piecewise functions, including handling endpoints correctly?
• Can you distinguish exponential growth vs decay and identify an exponential asymptote?
• Can you describe transformations from y=f(x) to a transformed equation?

Final Exam Pitfalls

1. Mistake: Treating f(x) as multiplication f\cdot x. Correct approach: read f(x) as a single output value—substitute the input into the rule.
2. Mistake: Ignoring domain constraints (especially denominator =0, log argument \le 0, or negative radicands). Correct approach: list restrictions first, then solve.
3. Mistake: Reversing horizontal shifts (e.g., thinking f(x-3) shifts left). Correct approach: inside shift is opposite—f(x-h) moves right by h.
4. Mistake: Dropping endpoints in piecewise problems. Correct approach: pay attention to
5. Mistake: Picking a matching graph using only one feature (like intercept) while ignoring end behavior/asymptotes. Correct approach: confirm at least two features (e.g., intercept + shape/end behavior).

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Claude Opus 4.6

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What You Need to Know

• Functions account for a significant portion of the ACT Math section — roughly 12–15% of questions fall under the "Functions" reporting category (approximately 7–9 out of 60 questions). Mastery here is high-yield.
• You must be comfortable with function notation, evaluating functions, interpreting graphs, and recognizing how transformations shift or stretch a graph. These skills appear repeatedly across difficulty levels.
• The ACT tests functions across multiple representations: equations, tables, graphs, and word problems. Being fluent in translating between these forms is essential.
• Know your parent functions (linear, quadratic, absolute value, square root, exponential, logarithmic) and how algebraic changes to the equation correspond to visual changes on the graph.

Function Definition and Notation

A function is a rule that assigns exactly one output to each input. If x is the input, then f(x) represents the output — read as "f of x."

Key Ideas

• Domain: the set of all valid inputs (x-values)
• Range: the set of all outputs (y-values)
• Vertical Line Test: a graph represents a function if and only if every vertical line crosses the graph at most once
• f(3) = 7 means "when the input is 3, the output is 7" — equivalently, the point (3, 7) is on the graph

Evaluating Functions

To evaluate f(a), substitute a for every x in the expression.

Example: If f(x) = 2x^2 - 3x + 1, find f(-2).

f(-2) = 2(-2)^2 - 3(-2) + 1 = 2(4) + 6 + 1 = 15

Composition of Functions

(f \circ g)(x) = f(g(x))

Evaluate the inner function first, then plug that result into the outer function.

Exam Focus

• Why it matters: Nearly every function question on the ACT requires correct use of notation. This is foundational.
• Typical question patterns:
  • Evaluate f(a) given an algebraic rule or a table of values
  • Find f(g(2)) using two function definitions
  • Determine domain restrictions (e.g., denominators ≠ 0, radicands ≥ 0)
• Common mistakes:
  • Confusing f(x + 2) with f(x) + 2 — the first substitutes x + 2 into the function; the second adds 2 to the output
  • Reversing the order in composition: f(g(x)) \neq g(f(x)) in general

Linear Functions

A linear function has the form:

f(x) = mx + b

where m is the slope (rate of change) and b is the y-intercept.

Slope formula between two points (x1, y1) and (x2, y2):

m = \frac{y2 - y1}{x2 - x1}

• Parallel lines have equal slopes: m1 = m2
• Perpendicular lines have negative reciprocal slopes: m1 \cdot m2 = -1

Exam Focus

• Why it matters: Linear functions are the most frequently tested function type on the ACT — they appear in pure math and in word-problem contexts.
• Typical question patterns:
  • Find the equation of a line through two points
  • Determine if two lines are parallel or perpendicular
  • Interpret slope as a rate of change in a real-world scenario
• Common mistakes:
  • Flipping the slope formula: always do y-difference over x-difference, not the reverse
  • Sign errors when working with negative slopes or negative coordinates

Polynomial Functions

Polynomial functions have the general form:

f(x) = anx^n + a{n-1}x^{n-1} + \cdots + a1x + a0

The degree (highest power of x) determines the function's general behavior.

Key Facts

• A polynomial of degree n has at most n real zeros (x-intercepts) and at most n - 1 turning points
• Quadratics (degree 2): f(x) = ax^2 + bx + c — parabola shape, vertex at x = -\frac{b}{2a}
• Zeros/roots: values of x where f(x) = 0. Factor or use the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
• End behavior: for large |x|, the leading term anx^n dominates. If n is even and an > 0, both ends go up; if n is odd and a_n > 0, left end goes down and right end goes up.

Exam Focus

• Why it matters: Quadratics dominate, but the ACT also tests higher-degree polynomial behavior — especially reading graphs.
• Typical question patterns:
  • Find the vertex or axis of symmetry of a parabola
  • Determine the number of real roots using the discriminant b^2 - 4ac
  • Identify end behavior from a given equation
• Common mistakes:
  • Forgetting the sign in the vertex formula — it's x = -\frac{b}{2a}, not \frac{b}{2a}
  • Miscounting zeros: a "touch" at the x-axis (double root) counts as one zero, not two distinct intercepts

Radical Functions

A radical function involves a root, most commonly:

f(x) = \sqrt{x} or more generally f(x) = \sqrt[n]{x}

Key Ideas

• Domain of f(x) = \sqrt{x}: x \geq 0 (the expression under an even root must be non-negative)
• The graph of y = \sqrt{x} starts at the origin and increases, curving to the right
• To solve radical equations, isolate the radical and then square both sides — always check for extraneous solutions

Example: Solve \sqrt{x + 3} = 5

Square both sides: x + 3 = 25, so x = 22. Check: \sqrt{25} = 5 ✓

Exam Focus

• Why it matters: The ACT frequently tests domain restrictions and solving radical equations.
• Typical question patterns:
  • "What is the domain of f(x) = \sqrt{2x - 6}?" → Solve 2x - 6 \geq 0 → x \geq 3
  • Solve a radical equation and identify extraneous solutions
• Common mistakes:
  • Forgetting to check solutions — squaring can introduce answers that don't satisfy the original equation
  • Allowing negative values under an even root when finding the domain

Piecewise Functions

A piecewise function is defined by different rules on different intervals of the domain:

f(x) = \begin{cases} x + 1 & \text{if } x < 0 \ x^2 & \text{if } x \geq 0 \end{cases}

To evaluate, determine which interval the input falls in, then use the corresponding rule.

Example: Find f(-3) and f(2) for the function above.

• f(-3): since -3 < 0, use x + 1 → f(-3) = -3 + 1 = -2
• f(2): since 2 \geq 0, use x^2 → f(2) = 4

Exam Focus

• Why it matters: Piecewise functions test careful reading and attention to inequality boundaries — a favorite for medium-difficulty ACT questions.
• Typical question patterns:
  • Evaluate a piecewise function at a specific value
  • Match a piecewise rule to a graph (look for breaks or changes in slope)
  • Determine continuity at the boundary point
• Common mistakes:
  • Using the wrong piece — pay close attention to whether the boundary uses < vs. \leq
  • Misreading the graph at transition points (open vs. closed circles)

Exponential and Logarithmic Functions

Exponential Functions

f(x) = a \cdot b^x

• If b > 1: exponential growth
• If 0 < b < 1: exponential decay
• The graph has a horizontal asymptote at y = 0 (for the basic form) and passes through (0, a)

Logarithmic Functions

A logarithm is the inverse of an exponential:

\log_b(x) = y \iff b^y = x

Key properties:

Memory Aid: "A logarithm answers the question: what exponent do I need?" — so \log_2(8) = 3 because 2^3 = 8.

Exam Focus

• Why it matters: Exponential and logarithmic questions appear at the medium-to-hard difficulty level and are high-value for boosting your score into the 30+ range.
• Typical question patterns:
  • Evaluate \log_3(81) or similar
  • Identify growth vs. decay from an equation or word problem
  • Use log properties to simplify or solve equations
• Common mistakes:
  • Confusing \logb(x + y) with \logb(x) + \log_b(y) — there is NO sum rule for logs of sums
  • Forgetting the domain restriction: \log_b(x) is only defined for x > 0

Function Transformations and Translations

Starting from a parent function f(x), transformations modify the graph predictably:

Memory Aid — "Horizontal is Backwards": Changes inside the function (applied to x) work in the opposite direction from what you might expect. f(x - 3) shifts right, not left.

Exam Focus

• Why it matters: Transformation questions are among the most predictable on the ACT — learn the table above and you can answer them quickly.
• Typical question patterns:
  • "The graph of y = f(x) is shifted 2 units left and 3 units up. What is the new equation?" → y = f(x + 2) + 3
  • Identify a transformation from two given graphs
  • Determine the vertex of a transformed quadratic y = a(x - h)^2 + k
• Common mistakes:
  • Getting horizontal shifts backwards — f(x - h) shifts right, f(x + h) shifts left
  • Confusing vertical stretch with horizontal stretch

Analyzing Graphs and Key Features of Functions

The ACT frequently asks you to read and interpret graphs. Key features to identify:

• Intercepts: x-intercepts (where f(x) = 0) and y-intercept (where x = 0)
• Increasing/Decreasing: a function is increasing where the graph goes up from left to right, decreasing where it goes down
• Maximum and Minimum values: the highest and lowest points — can be local (relative) or global (absolute)
• Symmetry: even functions satisfy f(-x) = f(x) (symmetric about the y-axis); odd functions satisfy f(-x) = -f(x) (symmetric about the origin)
• Asymptotes: lines the graph approaches but never reaches — vertical, horizontal, or oblique
• Zeros: the x-values where the function crosses or touches the x-axis
• Period (for periodic functions): the length of one complete cycle

Exam Focus

• Why it matters: Graph-reading questions span all difficulty levels. The ACT may show you a graph you've never seen and ask you to extract information — this tests understanding over memorization.
• Typical question patterns:
  • "Over what interval is the function decreasing?"
  • "What is the maximum value of f(x) on the interval [0, 5]?"
  • "How many real solutions does f(x) = 3 have?" (Count where y = 3 intersects the graph)
• Common mistakes:
  • Confusing the x-value of a maximum with the maximum value itself (the maximum value is the y-coordinate)
  • Misreading open vs. closed endpoints on a graph
  • Forgetting that "number of solutions to f(x) = k" means counting horizontal-line intersections

Quick Review Checklist

• ☐ Can you evaluate f(a) given an equation, table, or graph?
• ☐ Can you compute f(g(x)) for given functions f and g?
• ☐ Do you know how to find the slope of a line from two points and write the equation?
• ☐ Can you find the vertex of a quadratic using x = -\frac{b}{2a}?
• ☐ Can you determine the domain of a radical or logarithmic function?
• ☐ Can you evaluate a piecewise function at a boundary value?
• ☐ Do you know how to convert between exponential and logarithmic form?
• ☐ Can you describe how f(x - h) + k transforms the graph of f(x)?
• ☐ Can you identify intercepts, maxima, minima, and intervals of increase/decrease from a graph?
• ☐ Do you know the difference between even and odd functions?

Final Exam Pitfalls

1. Horizontal shift direction error: Students see f(x - 5) and shift left. Remember: subtracting inside the function shifts right. Always think "opposite."

2. Extraneous solutions with radicals: After squaring both sides of an equation, you must substitute back into the original to verify. The ACT will include the extraneous answer as a trap choice.

3. Log of a sum misconception: \log(a + b) \neq \log a + \log b. The product rule is \log(ab) = \log a + \log b. Do not distribute a log across addition.

4. Wrong piece in a piecewise function: Read the domain conditions carefully. If the boundary says x < 2 for one piece and x \geq 2 for another, f(2) uses the second piece. Watch the inequality symbols.

5. Confusing maximum value with its location: If a parabola's vertex is at (3, 7), the maximum value is 7 (the y-coordinate), not 3. The ACT answer choices will include both numbers.

6. Slope formula sign errors: When computing \frac{y2 - y1}{x2 - x1} with negative coordinates, use parentheses carefully. For points (2, -3) and (-1, 5): m = \frac{5 - (-3)}{-1 - 2} = \frac{8}{-3}$$. Rushing leads to dropped negatives.