Graphing Quadratics Basics
A free Algebra I lesson from the “Quadratic Foundations” unit, with a worked example and practice problems including step-by-step solutions.
The graph of a quadratic is a parabola. Its vertex is the turning point, its axis of symmetry cuts the graph in half, and its zeros are the x-intercepts.
What you'll learn
- Identify the vertex and axis of symmetry
- Determine whether a parabola opens up or down
- Connect zeros to x-intercepts
Worked example
Problem. For y = (x - 2)^2 + 3, identify the vertex.
- Vertex form is y = a(x - h)^2 + k.
- Here h = 2 and k = 3.
- The vertex is (2, 3).
Answer: (2, 3)
Practice problems
1. The graph of a quadratic is called a...
Choices: Parabola · Line · Circle · Histogram
Show solution
- Warm-up: First identify exactly what the question is asking: The graph of a quadratic is called a...
- For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
- Quadratic graphs are parabolas.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Parabola
2. For y = x^2, the vertex is...
Choices: (0, 0) · (1, 0) · (0, 1) · (-1, 1)
Show solution
- Warm-up: First identify exactly what the question is asking: For y = x^2, the vertex is...
- For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
- The parent quadratic turns at the origin.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: (0, 0)
3. Does y = 2x^2 open up or down?
Choices: Up · Down · Left · Right
Show solution
- Warm-up: First identify exactly what the question is asking: Does y = 2x^2 open up or down?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The coefficient of x^2 is positive.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Up
4. For y = -x^2 + 4, the parabola opens...
Choices: Down · Up · Right · Left
Show solution
- Core Practice: First identify exactly what the question is asking: For y = -x^2 + 4, the parabola opens...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The coefficient of x^2 is negative.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Down
5. For y = (x - 3)^2 - 2, the vertex is...
Choices: (3, -2) · (-3, -2) · (3, 2) · (-3, 2)
Show solution
- Core Practice: First identify exactly what the question is asking: For y = (x - 3)^2 - 2, the vertex is...
- For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
- h = 3 and k = -2.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: (3, -2)
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