A. Quadratic Functions · Quadratic Functions

From Vertex Form to Standard Form

A2.7About 65 minutes

Learning goals

I can square a binomial correctly.
I can write a quadratic function given in vertex form in standard form.
I can verify that two forms are equivalent with a table or graphing technology.
I can explain what each form tells me in a context.

Words to know

vertex form
f(x) = a(x − h)² + k; shows the vertex (h, k).

standard form
f(x) = ax² + bx + c; shows the y-intercept c.

expand
Multiply out brackets to write an expression as a sum of terms.

perfect square trinomial
A trinomial that is the square of a binomial, such as x² + 6x + 9 = (x + 3)².

equivalent
Giving the same output for every input; equivalent functions have the same graph.

Two ways to write the same function

A quadratic function can be written in more than one form. Two of them are:

Vertex form: f(x) = a(x − h)² + k. It shows the vertex (h, k) at a glance.
Standard form: f(x) = ax² + bx + c. It shows the y-intercept c at a glance, and it is the form most formulas use (including the quadratic formula in Quadratic Expressions and Equations).

In this lesson you will change vertex form into standard form by expanding and simplifying, and verify that the two forms really are the same function.

The key skill: squaring a binomial

To expand a(x − h)² + k, you must first square the binomial (x − h). Remember:

(x + p)² = x² + 2px + p²
(x − p)² = x² − 2px + p²

The middle term is twice the product of x and p. One of the most common errors in this unit is to write (x + 3)² = x² + 9. The area model below shows why that is wrong.

A square with side x + 3 is made of one x-by-x square, two x-by-3 rectangles and one 3-by-3 square. So (x + 3)² = x² + 6x + 9. Leaving out the two middle rectangles gives the wrong answer x² + 9.

The method: vertex form to standard form

Square the binomial: write (x − h)² as x² − 2hx + h².
Multiply every term of the trinomial by a (use brackets!).
Add k and collect like terms.

The order of operations matters: square first, then multiply by a, then add k.

Worked example

Example 1: f(x) = 2(x − 3)² − 5

Write f(x) = 2(x − 3)² − 5 in standard form.

Square: (x − 3)² = x² − 6x + 9.
Multiply by 2: 2(x² − 6x + 9) = 2x² − 12x + 18.
Add −5: 2x² − 12x + 18 − 5 = 2x² − 12x + 13.

f(x) = 2x² − 12x + 13.

Worked example

Example 2: A negative a

Write f(x) = −(x + 4)² + 7 in standard form.

Square: (x + 4)² = x² + 8x + 16.
Multiply by −1. The negative changes every sign in the bracket: −(x² + 8x + 16) = −x² − 8x − 16.
Add 7: −x² − 8x − 16 + 7 = −x² − 8x − 9.

f(x) = −x² − 8x − 9.

Verifying that the forms are equivalent

Two expressions are equivalent if they give the same output for every input. You can verify this in several ways.

With graphing technology: graph both equations in Desmos or on a graphing calculator. If they are equivalent, you see only one curve (the second one is drawn exactly on top of the first). In the table feature, the two columns of outputs match.
By hand: substitute a few x-values into both forms. Two different quadratics can agree at two points at most, so if they agree at three different x-values, the quadratics are identical. (This is a quick check, not a replacement for careful algebra.)

Worked example

Example 3: Expand and verify

Write f(x) = 0.5(x − 2)² + 1 in standard form. Verify with a table of values.

Square: (x − 2)² = x² − 4x + 4.
Multiply by 0.5: 0.5x² − 2x + 2. Add 1: f(x) = 0.5x² − 2x + 3.
Check x = 0: vertex form 0.5(4) + 1 = 3; standard form 0 − 0 + 3 = 3. ✓
Check x = 2: vertex form 0 + 1 = 1; standard form 2 − 4 + 3 = 1. ✓
Check x = 4: vertex form 0.5(4) + 1 = 3; standard form 8 − 8 + 3 = 3. ✓ Three matches, so the forms are equivalent.

f(x) = 0.5x² − 2x + 3.

Both equations produce exactly the same parabola. The vertex form shows the vertex (2, 1); the standard form shows the y-intercept 3.

Worked example

Example 4: Two forms, two stories

A ball's height in metres after t seconds is h(t) = −5(t − 2)² + 21. Write h in standard form. What does each form tell you about the ball?

Square: (t − 2)² = t² − 4t + 4.
Multiply by −5: −5t² + 20t − 20. Add 21: h(t) = −5t² + 20t + 1.
Vertex form: the vertex is (2, 21), so the ball reaches its maximum height of 21 m after 2 s.
Standard form: c = 1, so h(0) = 1: the ball was released 1 m above the ground.

h(t) = −5t² + 20t + 1. Maximum 21 m at 2 s; released from 1 m.

A useful pattern

Expanding the general vertex form gives

a(x − h)² + k = ax² − 2ahx + (ah² + k).

So a stays the same, b = −2ah and c = ah² + k. Since b = −2ah, the axis of symmetry is h = −b ÷ (2a). You will use this idea again in Sketching from Standard Form.

Just for fun

The mistake (a + b)² = a² + b² is so common that it has a nickname: the freshman's dream. Sadly, it only comes true when a or b is 0.

Hands-on activity

A paper proof that (x + 3)² is not x² + 9

You need: a sheet of paper or thin cardboard, a ruler, a pencil, and scissors (take care with scissors; a grown-up can help). A calculator is optional.

Let x = 10 cm. Draw a square 13 cm by 13 cm. Its area is (10 + 3)² = 169 cm².
Inside it, draw lines to split it into a 10 by 10 square, two 10 by 3 rectangles, and a 3 by 3 square.
Cut out the four pieces. Write each piece's area on it: 100, 30, 30, 9. Add them.
Now put only the 100 and 9 pieces together. How much area is missing compared with the 13 by 13 square?
Repeat with x = 6 cm and (x + 2)² if you have time. Write a sentence explaining why (x + p)² = x² + 2px + p².

What to look for: 100 + 30 + 30 + 9 = 169 cm², which matches 13². Using only x² + 9 = 109 misses 60 cm², the two 3x rectangles (2 × 3 × 10 = 60). For x = 6 and p = 2: 64 = 36 + 12 + 12 + 4.

Your turn

Write f(x) = (x − 4)² + 1 in standard form.

Write f(x) = 3(x + 1)² − 2 in standard form.

Write f(x) = −2(x − 5)² + 10 in standard form.

Write f(x) = 0.5(x + 6)² − 4 in standard form.

Verify that (x − 4)² + 1 and x² − 8x + 17 agree at x = 0, 4 and 5.

Find the y-intercept of f(x) = −(x − 3)² + 4 by writing it in standard form.

A student says (x − 2)² + 3 = x² + 7. Test x = 1, then give the correct standard form.

More practice: warm-up

Expand (x + 5)².

Expand (x − 1)².

Expand 2(x² + 3x + 1).

In f(x) = 4x² − 3x + 7, state a, b and c.

State the y-intercept of f(x) = x² + 2x − 8.

More practice: on your own

Write f(x) = −3(x − 2)² + 12 in standard form, then factor it to find the zeros.

Write f(x) = (1/4)(x + 8)² − 16 in standard form.

h(t) = −4.9(t − 1.5)² + 13.025 gives a ball's height in metres. Write it in standard form and explain what each form tells you.

Show algebraically that f(x) = 2(x − 1)² + 3 and g(x) = 2x² − 4x + 5 are the same function.

Which one is not equivalent to the others? A: (x + 2)² − 1 B: x² + 4x + 3 C: (x + 1)(x + 3) D: (x − 2)² − 1

Challenge

Expand a(x − h)² + k and show that h = −b ÷ (2a). Use it to find the axis of symmetry of f(x) = 3x² − 18x + 4.

A parabola has vertex (−1, 4) and y-intercept 1. Find its equation in vertex form and in standard form.

a(x − 2)² + k is equivalent to 2x² + bx + 11. Find a, b and k.

Check your understanding

Quiz

What is (x − 3)² + 2 in standard form?

Which expansion is correct?

What is the y-intercept of f(x) = 2(x − 1)² − 5?

What is −(x + 2)² in standard form?

How does graphing technology show that two equations are equivalent?

Which form shows the y-intercept directly?

For parents and teachers

Big idea: expectation A2.7 asks students to express a quadratic function in standard form f(x) = ax² + bx + c, given the vertex form f(x) = a(x − h)² + k, and to verify with graphing technology that the forms are equivalent.

How to help: ask your teen to check each answer by substituting x = 0 and one more value into both forms. If a device is available, have them graph both forms in Desmos and look for a single curve.

Common difficulties:

(x + 3)² = x² + 9 (missing the middle term). The paper activity is a strong fix.
Multiplying by a before squaring, or adding k before multiplying by a.
Sign errors when a is negative: the negative must reach every term.
Multiplying only the first term of the trinomial by a.