3.1 Linear and Quadratic Models
Key Takeaways
- Linear ACT Math models show a constant rate of change; slope is the rate and the y-intercept is the value when the input is 0.
- Quadratic models are best read by form: standard form shows the y-intercept, factored form shows zeros, and vertex form shows the turning point.
- For parabola questions, separate the x-value of a vertex from the y-value; the question may ask for time, input, height, output, maximum, or minimum.
- Use calculators to confirm arithmetic or graph shape, but choose the model from rates, intercepts, zeros, and vertex clues first.
Linear and Quadratic Models on ACT Math
ACT Math often tests functions through models instead of through isolated formulas. A question may give a graph, a short situation, a table of values, or an equation and ask for a rate, intercept, zero, maximum, minimum, or meaning in context. The official ACT Math description places algebra and functions inside Preparing for Higher Math, so these questions sit near the core of the section rather than at the edge.
The time pressure matters. With 45 questions in 50 minutes, you need a fast way to decide whether a model is linear, quadratic, or neither. A linear model has a constant first difference and a constant slope. A quadratic model has a constant second difference, a turning point, and usually two symmetric sides around a vertical axis.
Linear Models: Rate Plus Starting Value
A linear equation is built around y = mx + b. The slope m is the constant rate of change. The intercept b is the value when x = 0. On ACT-style items, those two pieces often translate directly into a real context: dollars per ticket plus a setup fee, gallons per minute plus an initial amount, or points gained per round plus a starting score.
If two points are given, compute slope first: m = (change in y)/(change in x). Then use either point to find b. Do not assume the y-intercept is one of the given y-values unless one point has x = 0.
Worked example: a line passes through (3, 11) and (7, 23). The slope is (23 - 11)/(7 - 3) = 12/4 = 3. Substitute (3, 11) into y = 3x + b: 11 = 9 + b, so b = 2. The model is y = 3x + 2, and the graph crosses the y-axis at 2.
| Feature | What to compute | ACT trap |
|---|---|---|
| Slope | change in y divided by change in x | Reversing rise and run |
| y-intercept | output when x = 0 | Treating any starting data row as x = 0 |
| x-intercept | input when y = 0 | Reporting y instead of x |
| Parallel line | same slope | Matching intercepts instead of slopes |
| Perpendicular line | negative reciprocal slope | Forgetting the sign change |
Quadratic Models: Zeros, Vertex, and Symmetry
A quadratic model has an x^2 term and graphs as a parabola. The most useful ACT forms are standard form ax^2 + bx + c, factored form a(x - r1)(x - r2), and vertex form a(x - h)^2 + k. The sign of a tells whether the graph opens up or down. The vertex is the minimum when a is positive and the maximum when a is negative.
Each form advertises different information. Standard form gives the y-intercept quickly: c. Factored form gives the zeros quickly: r1 and r2. Vertex form gives the turning point quickly: (h, k). A strong ACT move is to switch forms only when the question asks for information the current form does not show.
Worked example: f(x) = (x - 2)(x + 6). The zeros are x = 2 and x = -6. The axis of symmetry is halfway between them: (2 + -6)/2 = -2. To find the vertex value, evaluate f(-2): (-4)(4) = -16. Because the leading coefficient is positive, the vertex (-2, -16) is a minimum.
Choosing the Fast Method
Use this decision list before calculating:
- If the question asks for rate of change, slope, or a constant increase, look for a linear model.
- If the question asks for a maximum, minimum, turning point, or symmetric graph, look for a quadratic model.
- If a quadratic is already factored, read the zeros before expanding.
- If the answer choices are equations, match slope, intercept, zeros, or vertex before doing full algebra.
- If a graph is shown, check the scale on both axes before estimating.
A common trap is confusing average rate of change with instantaneous-looking language. ACT Math does not require calculus, so a rate over an interval usually means slope between two points. For a quadratic, the average rate from x = 1 to x = 5 is the slope of the secant line, not the slope at the vertex.
Another trap is stopping at an intermediate result. If the problem asks when the height is zero, solve for the x-values. If it asks for the maximum height, report the y-value at the vertex. If it asks for the time at which the maximum occurs, report the x-value of the vertex. The same parabola can support all three answers.
Calculator strategy should be selective. A graphing calculator can confirm the shape or zeros, but entering a messy expression often takes longer than factoring, completing a square, or checking the answer choices. On ACT Math, setup is the score-maker; the calculator should verify or reduce arithmetic, not replace the model choice.
Reading Tables Without Drawing the Whole Graph
Tables are a quick way to classify a model. If x increases by equal steps and y increases by equal amounts, the model is linear. If the first differences change but the second differences are constant, a quadratic model is likely. For example, y-values 5, 11, 21, 35 have first differences 6, 10, 14 and second differences 4, 4, so a quadratic pattern is present.
When answer choices contain equations, test a small x-value from the table before expanding everything. One substitution can eliminate several choices. Then use slope, intercept, zero, or vertex clues to finish. This is especially useful late in the section when algebraic forms look similar and time is limited.
A line passes through (2, 5) and (6, 13). Which equation represents the line?
For f(x) = (x - 3)^2 - 16, what are the zeros of the function?
The function g(x) = -2(x + 1)^2 + 8 models a parabola. Which statement is true?