A quadratic graphs as a parabola, a symmetric U. A positive leading coefficient opens upward with a minimum; negative opens downward with a maximum. The vertex sits at x = −b/2a, and the axis of symmetry runs through it. The y-intercept is c; the x-intercepts are the roots.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The axis of symmetry
x = 2 for y = x² − 4x + 3. Points at equal distance from it share a height: x = 0 and x = 4 both give 3; x = 1 and x = 3 both give 0. Find the vertex, then points come in pairs.
Intercepts
The y-intercept is c, at x = 0: (0, 3). The x-intercepts solve x² − 4x + 3 = 0: (x − 1)(x − 3) = 0, at 1 and 3. Their average, 2, is the vertex x: the roots are symmetric about the axis.
Two, one, or none
y = x² − 4x + 5 has vertex (2, 1), above the axis and opening up: no x-intercepts. y = x² − 4x + 4 has vertex (2, 0): one, a touch. The vertex height and the direction decide how many.
The size of a
y = 3x² is narrower than y = x²: at x = 1 it is already at 3. y = ½x² is wider. A larger |a| makes a narrower parabola; values between −1 and 1 widen it.
Sketching
Vertex (2, −1). Intercepts (0, 3), (1, 0), (3, 0). The mirror of (0, 3) is (4, 3). Five points, and the U through them. Everything is measured from the vertex.
Range
Opening upward from a vertex at −1, the parabola takes every y ≥ −1 and nothing below. The vertex y is the minimum value of the function.
Step 2: Try It Yourself
Tap and try it out.
Analyze y = −2x² + 8x − 6: direction, vertex, axis, intercepts, and a mirror point
- 1a = −2 < 0: opens downward, a maximumthe sign of a
Step 0 of 4
Step 3: In Real Life
A fountain’s arc
A fountain jet is a parabola. The vertex is its peak; the x-intercepts are where it leaves and lands. Designers read all three from the graph.
Step 4: Watch an Example
One step at a time.
Watch Rosa Locate a Vertex
Rosa analyses y = x² − 4x + 3.
- Step 1
The leading coefficient is 1, positive, so the parabola opens upward. The vertex sits at x = −b ÷ 2a = 4 ÷ 2 = 2.
Step 5: Your Turn
Practice makes it stick.
The Turning Point
Problem 1 of 2
y = x² − 4x + 3. What is the x-coordinate of the vertex?
The Intercept
Problem 2 of 2
y = x² − 4x + 3. What is the y-intercept?
Read the Parabola
1 of 8
y = x² − 6x + 5. Vertex x-coordinate?
2 of 8
y = 2x² + 8x. Vertex x-coordinate?
3 of 8
y = −x² + 4. Does it open up or down?
4 of 8
y = 3x² − 12x + 7. What is the y-intercept?
5 of 8
y = x² − 6x + 5 at x = 3. What is y?
6 of 8
A parabola opening upward has a maximum or minimum?
7 of 8
Sort each coefficient by what it controls.
Tap something to move it.
- Empty
- Empty
8 of 8
y = x² − 10x + 2. Vertex x-coordinate?
Step 6: Quick Check
Show what you know.
Question 1 of 2
y = x² − 8x + 1. What is the vertex x-coordinate?
Question 2 of 2
What does a negative leading coefficient mean?
What You Learned
- A quadratic graphs as a symmetric parabola.
- The vertex sits at x = −b ÷ 2a, and c is the y-intercept; the roots are symmetric about the axis.
- The sign of a decides the direction; its size decides the width.
- Find the vertex first; everything else comes in mirrored pairs.