Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

Key Features of a Parabola: Vertex, Axis of Symmetry, Zeros and Intercepts

1.3About 65 minutes

Learning goals

I can identify the vertex, axis of symmetry, y-intercept, zeros and maximum or minimum value of a parabola.
I can use the correct terminology and notation (point, number or equation) for each feature.
I can find the axis of symmetry halfway between the zeros or any two points at the same height.
I can tell whether a parabola has 0, 1 or 2 zeros.

Words to know

vertex
The turning point of a parabola: its lowest point (opens up) or highest point (opens down).

axis of symmetry
The vertical line through the vertex that divides the parabola into mirror-image halves; its equation is x = h.

minimum value
The smallest y-value of a parabola that opens up: the y-coordinate of its vertex.

maximum value
The largest y-value of a parabola that opens down: the y-coordinate of its vertex.

zeros
The x-values where a graph meets the x-axis (y = 0). Also called x-intercepts or roots.

y-intercept
The y-value where a graph crosses the y-axis (x = 0).

Talking about parabolas like a mathematician

If you describe a parabola as "a U that goes down to about there and then back up", nobody can draw it from your description. Mathematicians use a small set of key features. Once you know them, five numbers can describe a parabola almost completely. Today you will learn the names, how to find each feature on a graph or in a table, and how they are connected.

The key features

Vertex: the turning point of the parabola. It is the lowest point if the parabola opens up, or the highest point if it opens down. Write it as a point (x, y).
Axis of symmetry: the vertical line through the vertex that cuts the parabola into two mirror-image halves. Because it is a vertical line, its equation is x = (a number), the x-coordinate of the vertex.
Minimum or maximum value: the y-coordinate of the vertex. A parabola that opens up has a minimum value; one that opens down has a maximum value. This is a single number, not a point.
y-intercept: where the parabola crosses the y-axis (where x = 0). For y = ax² + bx + c, it is c.
Zeros (also called x-intercepts or roots): the x-values where the parabola meets the x-axis (where y = 0). A parabola can have 2, 1 or 0 zeros.
Direction of opening: up (a > 0) or down (a < 0).

All the key features of y = x² − 6x + 5. The vertex (3, −4) lies on the axis of symmetry x = 3, exactly halfway between the zeros 1 and 5. The minimum value is −4.

Worked example

Example 1: Reading features from a graph

Use the graph of y = x² − 6x + 5 above to state each key feature.

Direction: it opens up (a = 1 > 0).
Vertex: the lowest point is (3, −4).
Axis of symmetry: the vertical line through the vertex: x = 3.
Minimum value: the y-coordinate of the vertex: −4.
y-intercept: the graph crosses the y-axis at 5 (check: at x = 0, y = 0 − 0 + 5 = 5).
Zeros: the graph meets the x-axis at x = 1 and x = 5 (check: 1 − 6 + 5 = 0 and 25 − 30 + 5 = 0).

Opens up; vertex (3, −4); axis x = 3; minimum value −4; y-intercept 5; zeros 1 and 5.

Two ways to say it

"The vertex is (3, −4)" is a point. "The minimum value is −4" is a number: the smallest y the relation ever reaches. "The axis of symmetry is x = 3" is a line (an equation). Keep the three kinds of answers straight.

Worked example

Example 2: Reading features from a table

For y = −x² + 4x + 5, make a table for x = −1 to 5 and state the key features without drawing the graph.

Values: x = −1: −1 − 4 + 5 = 0; x = 0: 5; x = 1: −1 + 4 + 5 = 8; x = 2: −4 + 8 + 5 = 9; x = 3: −9 + 12 + 5 = 8; x = 4: −16 + 16 + 5 = 5; x = 5: −25 + 20 + 5 = 0.
The y-values rise to 9 and then fall in mirror order (8, 5, 0 on each side of 9). So the axis of symmetry is x = 2 and the vertex is (2, 9).
a = −1 < 0, so it opens down and 9 is the maximum value.
y = 0 at x = −1 and x = 5, so the zeros are −1 and 5. The y-intercept is 5.

Opens down; vertex (2, 9); axis x = 2; maximum value 9; zeros −1 and 5; y-intercept 5.

The axis of symmetry is always halfway

Because a parabola is symmetric, any two points at the same height are the same distance from the axis of symmetry, one on each side. So the axis is exactly halfway between them. The two zeros are one such pair.

Axis of symmetry: x = (x₁ + x₂) ÷ 2, where x₁ and x₂ are the x-coordinates of two points with the same y-value (for example, the two zeros).

Once you know the axis, substitute that x into the equation to find the y-coordinate of the vertex.

Points (−1, 6) and (7, 6) are at the same height on a parabola. The axis of symmetry is halfway between −1 and 7: (−1 + 7) ÷ 2 = 3, so it is x = 3.

Worked example

Example 3: Vertex from the zeros

A parabola has zeros −3 and 5 and its equation is y = 2x² − 4x − 30. Find the axis of symmetry, the vertex and the minimum value.

Axis of symmetry: halfway between the zeros: x = (−3 + 5) ÷ 2 = 2 ÷ 2 = 1.
The vertex is on this line, so substitute x = 1: y = 2(1)² − 4(1) − 30 = 2 − 4 − 30 = −32.
Check one zero: 2(5)² − 4(5) − 30 = 50 − 20 − 30 = 0. ✓
a = 2 > 0, so it opens up and −32 is a minimum.

Axis x = 1; vertex (1, −32); minimum value −32.

A parabola that opens up can cross the x-axis twice, touch it once (at its vertex), or miss it completely. The same is true for parabolas that open down.

Connection

Finding the zeros from the equation (without a graph or table) means solving ax² + bx + c = 0. You will learn to do that by factoring and by other methods in the unit Quadratic Expressions and Equations. In this unit you will read zeros from graphs and tables, or use symmetry.

Just for fun

Why did the parabola break up with the straight line?

They were heading in different directions, and the parabola needed someone who could handle its ups and downs. (Also, it had a lot of vertex-ed emotions.)

Hands-on activity

Fold-and-find parabola

You need: graph paper (or squared paper printed from the internet), a pencil, and a window or light table for tracing.

Graph y = x² − 6x + 5 for x = −1 to 7 carefully on graph paper (make a table of values first; for example x = −1 and x = 7 both give y = 12).
Fold the paper along a vertical grid line so that the two halves of the parabola land exactly on top of each other. Hold it up to a window to check.
The fold is the axis of symmetry. Write its equation on the paper.
Mark the vertex, the zeros and the y-intercept and label each with its correct name.
Repeat with y = −x² + 4x + 5 from Example 2.

What to look for: the first fold is along x = 3 and the second along x = 2. The vertex sits right on the fold, and the two zeros land on top of each other when folded.

Your turn

A parabola has vertex (−2, 7) and opens down. State the axis of symmetry and the maximum or minimum value.

A parabola has zeros 2 and 10. What is its axis of symmetry?

For y = x² − 2x − 8, the zeros are −2 and 4. Find the vertex and the minimum value.

State the y-intercept of y = −3x² + 7x − 12.

A table shows x = 0, 1, 2, 3, 4, 5, 6 and y = 10, 4, 0, −2, −2, 0, 4. Find the zeros and the axis of symmetry.

Sketch or describe a parabola that opens up and has no zeros.

More practice: warm-up

What is the vertex of a parabola?

Is the axis of symmetry a horizontal or a vertical line?

A parabola has vertex (4, −1). What is the equation of its axis of symmetry?

What is another name for the zeros of a parabola?

Find the axis of symmetry if the zeros are −6 and 0.

More practice: on your own

For y = −2x² + 8x, the zeros are 0 and 4. Find the vertex, the maximum value and the y-intercept.

The points (−5, 12) and (1, 12) are on a parabola. Find its axis of symmetry.

A parabola has vertex (3, 5) and passes through (0, −4). Name another point on it without finding the equation.

A parabola opens down and has vertex (1, −2). How many zeros does it have? Explain.

For y = x² + 6x + 9, the only zero is −3. Describe the vertex and explain why there is only one zero.

Challenge

A parabola has a zero at x = −1 and its axis of symmetry is x = 4. Find the other zero.

The table of a parabola has y = 7 at x = 1 and y = 7 at x = 9, and y = −9 at x = 5. Is (5, −9) the vertex? Explain, and give the minimum or maximum value.

For y = x² − 10x + c, the axis of symmetry is x = 5. Find c if the minimum value is 3.

Check your understanding

Quiz

A parabola has vertex (−1, 6) and opens down. Its maximum value is…

The zeros of a parabola are −4 and 2. The axis of symmetry is…

Which feature is always found by setting x = 0?

How many zeros does y = x² + 4 have?

Points (2, 5) and (8, 5) lie on a parabola. The vertex must have x-coordinate…

Which describes y = −x² + 6x − 5 (zeros 1 and 5)?

For parents and teachers

Big idea: Expectation 1.3 asks students to identify the key features of a parabola (axis of symmetry equation, vertex coordinates, y-intercept, zeros, maximum or minimum value) and to use the right terms. Symmetry is the big tool: the axis is always halfway between two points at the same height.

How to help: Ask your teen to teach you the vocabulary with a sketch. Then quiz them: "Is that a point, a number or a line?" Folding a graph in half is a memorable way to see the axis of symmetry.

Common difficulties:

Writing the axis as y = 3 or as a point. It is a vertical line: x = 3.
Giving the vertex when asked for the maximum value (or the other way round).
Assuming every parabola has two zeros.
Arithmetic slips when finding the y-coordinate of the vertex, especially with negative x-values.