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

Vertex Form y = a(x − h)² + k: The Roles of a, h and k

2.2About 70 minutes

Learning goals

I can explain the roles of a, h and k in y = a(x − h)² + k using the correct terminology.
I can state the vertex, axis of symmetry, direction of opening and maximum or minimum value from vertex form.
I can find the y-intercept and the number of zeros from vertex form.
I can use the mapping rule (x, y) → (x + h, ay + k).

Words to know

vertex form
The form y = a(x − h)² + k of a quadratic relation; its vertex is (h, k).

parameter
A number in an equation, such as a, h or k, that can be changed to give a family of graphs.

vertical stretch / compression
Multiplying every y-value by a; a stretch if the size of a is greater than 1, a compression if it is between 0 and 1.

reflection in the x-axis
The flip given by a negative a; the parabola opens down.

horizontal / vertical translation
The slides given by h (right if positive) and k (up if positive).

Putting the three moves together

You now know what each of a, h and k does on its own. The vertex form of a quadratic relation uses all three at once:

y = a(x − h)² + k

Its big advantage is that you can read the vertex, the axis of symmetry, the direction of opening and the shape straight from the equation, with no table and no calculation. In this lesson you will explain the role of each parameter using the correct words.

The roles of the three parameters in vertex form. The vertex is (h, k), the axis of symmetry is x = h, and a controls the shape and direction.

Reading vertex form

For y = a(x − h)² + k:

Vertex: (h, k). Careful with h: in y = (x + 4)², h = −4.
Axis of symmetry: x = h.
Direction: up if a > 0, down if a < 0.
Maximum or minimum value: k (a minimum if a > 0, a maximum if a < 0).
Shape: vertical stretch by a factor of a if the size of a is greater than 1; vertical compression if it is between 0 and 1.
y-intercept: substitute x = 0: y = a(0 − h)² + k = ah² + k.
Number of zeros: compare the direction with the position of the vertex. Opens up with the vertex below the x-axis, or opens down with the vertex above it: 2 zeros. Vertex on the x-axis (k = 0): 1 zero. Otherwise: 0 zeros.

Mapping rule: every point (x, y) on y = x² moves to (x + h, ay + k). Multiply the y-coordinate by a first (stretch, compress, reflect), then translate.

Worked example

Example 1: Describe y = −3(x + 4)² + 1

Describe the transformations of y = x², and state the vertex, axis of symmetry, maximum or minimum value, y-intercept and number of zeros.

Match to y = a(x − h)² + k: a = −3, x + 4 = x − (−4) so h = −4, and k = 1.
a = −3: a reflection in the x-axis and a vertical stretch by a factor of 3.
h = −4: a translation 4 units left. k = 1: a translation 1 unit up.
Vertex (−4, 1); axis of symmetry x = −4; it opens down, so 1 is the maximum value.
y-intercept: −3(0 + 4)² + 1 = −3(16) + 1 = −47.
It opens down and its highest point (−4, 1) is above the x-axis, so it has 2 zeros.

Reflection in the x-axis, vertical stretch by 3, translation 4 left and 1 up. Vertex (−4, 1); axis x = −4; maximum value 1; y-intercept −47; 2 zeros.

Worked example

Example 2: All the features of y = 2(x − 3)² − 8

Find the key features of y = 2(x − 3)² − 8 and check them with the mapping rule.

a = 2, h = 3, k = −8: vertical stretch by 2, translation 3 right and 8 down.
Vertex (3, −8); axis x = 3; opens up; minimum value −8.
y-intercept: 2(0 − 3)² − 8 = 18 − 8 = 10.
Mapping (x, y) → (x + 3, 2y − 8): (0, 0) → (3, −8); (1, 1) → (4, −6); (−1, 1) → (2, −6); (2, 4) → (5, 0); (−2, 4) → (1, 0).
So the zeros are 1 and 5, and they are symmetric about x = 3, as they should be.

Vertex (3, −8), axis x = 3, minimum −8, y-intercept 10, zeros 1 and 5.

y = 2(x − 3)² − 8 compared with y = x² (dashed). It is narrower (stretched by 2) and its vertex has moved from (0, 0) to (3, −8).

Worked example

Example 3: From a description to an equation

A parabola has vertex (−2, 5), opens down, and is a vertical compression of y = x² by a factor of 1/2. Write its equation and say how many zeros it has.

Opens down and compressed by 1/2, so a = −1/2.
Vertex (−2, 5) gives h = −2 and k = 5.
y = −½(x − (−2))² + 5 = −½(x + 2)² + 5.
It opens down with its highest point above the x-axis, so it has 2 zeros.

y = −½(x + 2)² + 5, with 2 zeros.

Vertex form and standard form

Every vertex-form equation can be expanded into standard form. For example y = 2(x − 3)² − 8 = 2(x² − 6x + 9) − 8 = 2x² − 12x + 10. Notice a = 2 in both forms and c = 10 is the y-intercept you found. Expanding, and going the other way (from standard form to vertex form by completing the square), are both practised in the unit Quadratic Expressions and Equations.

Just for fun

Why is vertex form the most honest way to write a parabola?

Because it tells you right away where it stands: (h, k).

Hands-on activity

Parabola matching cards

You need: 12 small pieces of paper or index cards and a pencil.

On 6 cards write these equations: y = (x − 2)² + 1, y = −(x − 2)² + 1, y = 2(x + 1)² − 3, y = 0.5(x + 1)² − 3, y = −3x² + 4, y = (x + 3)².
On the other 6 cards write these descriptions, in a mixed-up order: "vertex (2, 1), opens down"; "vertex (−3, 0), touches the x-axis once"; "vertex (−1, −3), narrower than y = x²"; "vertex (2, 1), minimum value 1"; "vertex (0, 4), maximum value 4"; "vertex (−1, −3), wider than y = x²".
Shuffle all 12 cards face down and play a memory game with a family member: turn over two cards; keep them if the equation matches the description. Explain each match out loud using the words stretch, compression, reflection and translation.

What to look for (answer key): y = (x − 2)² + 1 ↔ vertex (2, 1), minimum value 1; y = −(x − 2)² + 1 ↔ vertex (2, 1), opens down; y = 2(x + 1)² − 3 ↔ vertex (−1, −3), narrower; y = 0.5(x + 1)² − 3 ↔ vertex (−1, −3), wider; y = −3x² + 4 ↔ vertex (0, 4), maximum value 4; y = (x + 3)² ↔ vertex (−3, 0), touches the x-axis once.

Your turn

State the vertex and axis of symmetry of y = 4(x − 1)² + 7.

Describe the transformations of y = x² that give y = −(x − 6)² − 2.

Is k a maximum or a minimum for y = 0.3(x + 5)² − 9? What is it?

Find the y-intercept of y = −2(x − 1)² + 6.

How many zeros does y = 5(x + 2)² + 3 have? Explain.

Write the equation of the parabola congruent to y = x², opening down, with vertex (4, −1).

More practice: warm-up

In y = a(x − h)² + k, which parameter moves the graph up or down?

What is the vertex of y = (x + 7)² − 3?

Which way does y = −0.5(x − 1)² + 2 open?

What is the axis of symmetry of y = 3(x + 2)² + 8?

Write the mapping rule for y = 2(x − 1)² + 5.

More practice: on your own

Use the mapping rule to find the images of (0, 0), (1, 1) and (2, 4) under y = −2(x + 3)² + 4.

Find the zeros of y = (x − 1)² − 16 using the mapping rule.

A parabola is y = x² stretched vertically by a factor of 4, reflected in the x-axis, translated 3 left and 10 up. Write its equation and its maximum value.

Two parabolas, y = 2(x − 5)² + k and y = −(x − 5)² + 3, have the same vertex. Find k.

Explain why y = a(x − h)² + k with a > 0 and k > 0 has no zeros.

Challenge

Expand y = −(x + 2)² + 9 into standard form and find its zeros from the vertex form. (Hint: when is (x + 2)² = 9?)

For y = a(x − 4)² + 6, the y-intercept is −10. Find a and describe the parabola.

A parabola has vertex (h, k) and passes through the origin. Show that its y-intercept is 0 and write a in terms of h and k (h ≠ 0).

Check your understanding

Quiz

What is the vertex of y = −5(x + 1)² + 8?

In y = a(x − h)² + k, which parameter tells you whether the parabola opens up or down?

Which describes y = 0.25(x − 2)² − 6?

The axis of symmetry of y = 3(x − 7)² + 2 is…

What is the y-intercept of y = 2(x + 3)² − 5?

How many zeros does y = −2(x − 4)² − 1 have?

Under y = 3(x − 1)² + 2, the point (2, 4) on y = x² maps to…

For parents and teachers

Big idea: Expectation 2.2 asks students to explain the roles of a, h and k in y = a(x − h)² + k using correct terminology (vertical stretch or compression, reflection in the x-axis, horizontal and vertical translation), and to identify the vertex and the axis of symmetry.

How to help: Play the matching-card game together. Ask your teen to describe a parabola using all three words: "It is reflected, stretched by 3, moved 4 left and 1 up." Then check in Desmos with sliders for a, h and k.

Common difficulties:

The sign of h: y = (x + 4)² has vertex x = −4.
Calling k the vertex instead of the y-coordinate of the vertex (or the max/min value).
Forgetting to multiply by a before adding k when using the mapping rule.
Counting zeros without considering both the direction and the vertex height.