Quadratic Functions: Graphs, Zeros, and the Vertex
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This is a paper math packet about quadratic functions — functions whose graphs are parabolas (U-shaped curves). You will read a quadratic written in standard form y = ax² + bx + c, find its vertex (the turning point) and its axis of symmetry, find its zeros (roots) by factoring, and interpret the vertex as a maximum or minimum in a real situation. Work by yourself with a pencil. A calculator is not required, but a graphing or scientific calculator is allowed if you have one. Show your work — every answer should show the numbers and steps you used, not just a final number. Everything you need (formulas, a worked example, a graph) is printed right here. If a question is hard, skip it, keep going, and come back. Circle or box your final answers.
Start Warm-Up: Retrieval 6 min
Bring back what you already know about evaluating and factoring.
1Evaluate a quadratic. A function is defined by f(x) = x² − 4x + 3. Find f(0) and f(2). Show the squaring, the multiplication, and the addition you used. (Remember: square the input first, then multiply, then combine.)
2Factor and use the Zero Product Property. Factor x² + 5x + 6 into two binomials (find two numbers that multiply to 6 and add to 5). Then find the two values of x that make the product equal 0.
Build Study the Math 8–12 min
What you need to know (with formulas)
A quadratic function has a graph that is a parabola — a symmetric U-shaped curve. Its standard form is:
y = ax² + bx + c (a ≠ 0)
The number a controls the opening. If a > 0 the parabola opens up and its vertex is the lowest point (a minimum). If a < 0 it opens down and its vertex is the highest point (a maximum). The number c is the y-intercept — the output when x = 0, the point (0, c).
Vertex. The vertex is the turning point of the parabola. Its x-coordinate is found from a and b:
xvertex = −b ÷ (2a)
Then substitute that x back into the function to get the y-coordinate. The vertex is the point (xvertex, yvertex).
Axis of symmetry. The parabola is a mirror image across the vertical line through the vertex:
x = −b ÷ (2a) (a vertical line)
Two points with the same output sit at the same distance on each side of this line. That is why the two zeros (below) are equally spaced around the axis.
Zeros (roots). The zeros (also called roots or x-intercepts) are the x-values where the output is 0 — where the parabola crosses the x-axis. To find them by factoring, set y = 0 and factor:
ax² + bx + c = 0 → (x − r₁)(x − r₂) = 0 → x = r₁ or x = r₂
This uses the Zero Product Property: if two factors multiply to 0, then at least one factor must be 0. Set each factor equal to 0 and solve.
Vertex as max or min in context. In a real story, the vertex often answers "how high" or "how much." For a thrown object, the vertex is the greatest height (a maximum): the x-coordinate tells when and the y-coordinate tells how high. For an area or cost problem the vertex can be a maximum area or a minimum cost.
Tip: the axis of symmetry x = −b ÷ (2a) and the vertex share the same x-value. Find x first; the axis is that vertical line and the vertex is that point on the curve.
Worked Example — y = x² − 4x + 3
Here a = 1, b = −4, c = 3. Because a = 1 > 0, the parabola opens up, so the vertex is a minimum. The y-intercept is c = 3, the point (0, 3).
Axis of symmetry / vertex x: x = −b ÷ (2a) = −(−4) ÷ (2·1) = 4 ÷ 2 = 2. So the axis of symmetry is the vertical line x = 2.
Vertex y: substitute x = 2: f(2) = (2)² − 4(2) + 3 = 4 − 8 + 3 = −1. The vertex is (2, −1) — the lowest point.
Zeros by factoring: set x² − 4x + 3 = 0. Two numbers that multiply to +3 and add to −4 are −1 and −3, so it factors as (x − 1)(x − 3) = 0. By the Zero Product Property, x − 1 = 0 or x − 3 = 0, giving x = 1 or x = 3. The parabola crosses the x-axis at (1, 0) and (3, 0).
Symmetry check: the two zeros 1 and 3 are the same distance (1 unit) on each side of the axis x = 2. The midpoint of the zeros, (1 + 3) ÷ 2 = 2, is the axis. ✔
| x | f(x) | Note |
|---|---|---|
| 0 | 3 | y-intercept (0, 3) |
| 1 | 0 | zero / x-intercept |
| 2 | −1 | vertex (minimum) |
| 3 | 0 | zero / x-intercept |
| 4 | 3 | mirror of (0, 3) |
3Using the worked example graph, find f(4) for f(x) = x² − 4x + 3, and explain in one sentence why f(4) has the same value as f(0). (Hint: use the axis of symmetry x = 2.)
Apply Use What You Know 20–26 min
Show the numbers you use in every item. A calculator is allowed.
4Read a graph. A parabola is described below. It opens up and passes through the points (−1, 0), (0, −3), (1, −4), (2, −3), and (3, 0). Use the description to answer 4a–4d.
| x | y |
|---|---|
| −1 | 0 |
| 0 | −3 |
| 1 | −4 |
| 2 | −3 |
| 3 | 0 |
4a. Name the two zeros (x-intercepts) — the x-values where y = 0.
4b. Give the vertex (the lowest point in the table) as an ordered pair, and state whether it is a maximum or a minimum.
4c. Give the axis of symmetry as an equation x = ___. Explain in one sentence how the two zeros show you the axis.
4d. What is the y-intercept (the value of y when x = 0)?
5Factor to find the roots. Solve x² − 7x + 10 = 0 by factoring. Show the two numbers you found (they multiply to 10 and add to −7), the factored form, and each root from the Zero Product Property. Then state the axis of symmetry (halfway between the roots).
6Match a quadratic to a context. A soccer ball is kicked and its height in feet after t seconds is h(t) = −16t² + 32t (here a = −16, b = 32, c = 0).
6a. Does this parabola open up or down? Is the vertex a maximum or a minimum? Answer in one sentence, using the sign of a.
6b. Find the time of the vertex using t = −b ÷ (2a), then find the greatest height by evaluating h at that time. Show your steps and interpret the vertex in the story (when is the ball highest, and how high?).
7Error analysis (a sign error in factoring). A student solved x² − x − 6 = 0 and wrote the work shown. Find the mistake, explain it, and give the correct roots.
Student's work:
x² − x − 6 = (x − 2)(x − 3) = 0
So x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.
What is wrong, and what are the correct roots? Fix it in the box.
Block Extra Applied Task block only · ~12–15 min
8Vertex form and a second model. The vertex form of a quadratic is y = a(x − h)² + k, where the vertex is the point (h, k) and the axis of symmetry is x = h. This form shows the vertex directly, with no calculation needed.
8a. A parabola is given by y = 2(x − 3)² − 5. Write its vertex (h, k) and its axis of symmetry. Does it open up or down, and is the vertex a max or a min?
8b. Area modeling. A rancher has 40 feet of fence to enclose a rectangular pen against a barn wall (the wall is one side, so the fence covers the other three sides). If the width is x, the two widths and one length use 2x + length = 40, so length = 40 − 2x, and the area is A(x) = x(40 − 2x) = −2x² + 40x. Find the width x at the vertex (use x = −b ÷ (2a)), then the maximum area. Show your steps and interpret the vertex (what width gives the biggest pen, and how big is it?).
Explain Justify Your Strategy 6–10 min
Question 9. A classmate says, "For the quadratic y = x² − 6x + 8, the vertex is at x = 4 because the two roots are 2 and 4." Is the classmate right? Find the true roots and the true vertex, and decide. Write a Claim (your answer), Evidence (the roots you found by factoring and the vertex x from −b ÷ (2a)), and Reasoning (why the axis of symmetry sits halfway between the roots, not at a root).
Sentence stems you may use: "The classmate is ___ because…" · "Factoring x² − 6x + 8 gives… so the roots are…" · "The axis of symmetry is x = −b ÷ (2a) = …" · "The vertex sits halfway between the roots because…" · "A root is where y = 0, but the vertex is where the curve turns, so…"
Close ACE Wrap-Up 5 min
In your own words, explain the difference between a zero (root) of a quadratic and its vertex. Which one is where the curve crosses the x-axis, and which one is where the curve turns?
Point to one item in this packet where you found a vertex and used it as a maximum or minimum in a real context. Give its number.
Give a new real-life example where finding a vertex answers a useful question (a maximum or a minimum) that was not in this packet.
Continue Early Finisher optional
Build your own quadratic. On the back of this page: (1) choose two whole-number roots you like (for example 2 and 5) and write a factored quadratic (x − r₁)(x − r₂); (2) multiply it out into standard form y = x² + bx + c; (3) find the axis of symmetry and the vertex; and (4) make a small table of 5 outputs and sketch the parabola, marking the vertex, the axis, and both zeros. Check that your two roots are the same distance on each side of your axis.