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: Picture It
Start with something you already know.
A ball thrown straight up
- You throw a ball straight up into the air.
- It rises fast, then slower, and stops for an instant at the top.
- Then it falls, slowly at first, then faster.
- One second before the top and one second after, it is at the same height.
- The trip down is the trip up played backward.
The ball’s height over time is a parabola. The top is its vertex.
The top comes at 2 seconds, and that moment is the axis of symmetry: heights come in pairs, equally early and equally late.
Where the picture stops: a thrown ball always gives a cap, opening downward. A positive a gives the same shape turned over, a valley with a lowest point.
A ball thrown straight up at 20 meters a second, its height each second, rounding gravity’s pull to 10 meters per second each second to keep the numbers simple.
| Seconds after the throw | Height in meters |
|---|---|
| 0 | 0 |
| 1 | 15 |
| 2 | 20 |
| 3 | 15 |
| 4 | 0 |
The top is at 2 seconds. The heights at 1 and 3 match, and so do 0 and 4: each pair is equally far from the top.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Work out y = x² − 6x + 8 for x = 0, 1, 2, 3, 4, 5 and 6. At which x is y the lowest?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Dev Sketch y = x² + 2x − 3
Dev reads y = x² + 2x − 3 from its equation: the direction, the vertex, the axis, the intercepts and a mirror point, with a check at the end.
- Step 1
The leading coefficient a is 1, which is positive, so the parabola opens upward. Its vertex will be its lowest point.
Watch Ana Read y = −x² + 6x − 5
A parabola that opens downward. Ana finds the direction and the axis, and you find the vertex height, a mirror point, and how many times it crosses the x-axis.
- Step 1
The leading coefficient is −1, which is negative, so the parabola opens downward. Its vertex will be its highest point.
Step 6: Your Turn
Practice makes it stick.
Where the Water Lands
Problem 1 of 15
A garden fountain’s jet follows h = −0.5x² + 4x, where x is the distance in feet from the nozzle and h is the height in feet. The water leaves the nozzle at x = 0. How far from the nozzle does it land?
feet
The Bake Sale Price
Problem 2 of 15
A bake-sale table’s profit in dollars is P = −2x² + 24x − 40, where x is the price of one box of cookies in dollars. What is the greatest profit the table can make, in dollars?
dollars
Read the Parabola
The sign of a first, then the vertex. Everything else is measured from it.
Problem 3 of 15
y = x² + 6x + 5. What is the y-coordinate of the vertex?
Problem 4 of 15
y = −2x² + 4x + 1. Which way does it open, and where is its vertex?
Problem 5 of 15
The point (5, 12) is on y = x² − 4x + 7. Another point on the curve also has height 12. What is its x-coordinate?
Problem 6 of 15
y = (x − 4)(x + 2) crosses the x-axis at x = 4 and x = −2. It opens upward. What is its lowest value of y?
Problem 7 of 15
y = −x² + 2x + 3. What is its range?
Whose Step?
One line in each page is a slip. Find it, then fix it.
Problem 8 of 15
Sam found the vertex of y = x² + 4x + 1. One line is a slip. Which?
Sam’s vertex of y = x² + 4x + 1
- 1a = 1, b = 4, c = 1
- 2x = −b/2a = 4/2 = 2
- 3y = 4 + 8 + 1 = 13
- Answervertex (2, 13)
Problem 9 of 15
What is the y-coordinate of the vertex of y = x² + 4x + 1, correctly?
Problem 10 of 15
Leo started a sketch of y = x² − 10x + 5 and placed a mirror point. One line is a slip. Which?
Leo’s points for y = x² − 10x + 5
- 1x = 10/2 = 5
- 2y = 25 − 50 + 5 = −20: vertex (5, −20)
- 3y-intercept (0, 5)
- Answermirror of (0, 5) is (5, 5)
Problem 11 of 15
What is the x-coordinate of the mirror of (0, 5) on y = x² − 10x + 5?
Pick the Move
Decide what to look at first. No need to work it all the way through.
Problem 12 of 15
Ana wants to know how many x-intercepts y = x² − 2x + 6 has, without solving anything. Which two facts does she need?
Problem 13 of 15
Maya knows y = x² − 10x + 21 crosses the x-axis at x = 3. What is the quickest way to the other crossing?
Same Axis? Same Width?
Judge from the coefficients, without plotting.
Problem 14 of 15
Select every parabola whose axis of symmetry is x = 2.
Problem 15 of 15
Select every parabola that has the same width as y = 2x².
Step 7: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
For which number b does y = x² + bx + 7 have its axis of symmetry at x = 4?
Stretch 2 of 2
For which number c does y = x² − 6x + c touch the x-axis at exactly one point?
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.
For the grown-up
The misconception this lesson targets is losing the minus sign in x = −b ÷ 2a, so that y = x² + 4x + 1 gets a vertex at x = 2 instead of −2. Its cousin is taking c, the y-intercept, for the vertex, because both are “the number you can read off”. One quick test settles either doubt: two x values equally far either side of a true axis give the same y, and on a false axis they do not.