Algebra II Quadratic Functions Review

Quadratics Review

04/01/2025

Key vocabulary

• Equation forms (standard, vertex, factored)

• Shape (parabola)

• Vertex

• Axis of symmetry

• Roots/Zeros/x-intercepts

• Concavity

Standard form

​f(x) = ax² + bx + c

This is the most usual form you will find a quadratic equation in. This orders the component terms from highest order to lowest.

a, b, & c are "coefficients," with a being considered the "leading coefficient."

Dropdown

Identify the coefficients in the following quadratic function:

​ f(x) = 7x² - 5x - 2

a = ​

Select answer

b = ​

Select answer

c = ​

Select answer

Vertex form

​f(x) = a(x - h)² + k

This form gives the vertex of the equation through h & k, where h is the x-coordinate of the vertex and k is the y-coordinate.

The equation
f(x) = 2(x - 3)² + 7
would have a vertex of (h, k), which would be at the point (3, 7).

Fill in the Blank

Identify the vertex of the following quadratic function in vertex form:

f(x) = 2(x + 1)² + 3

Type as a coordinate pair: (h, k)

(

-

,

)

Factored form

​f(x) = a(x - r₁)(x - r₂)

This form is made up of the factors that multiply together to get standard form. Remember your FOIL and Box Methods!

Notice that both terms are (x - r) -- that's because the roots ("r") have opposite signs from the factors of the standard equation.

Hotspot

Identify the roots of the graph:

Multiple Choice

What is the name for the shape of the graph of a quadratic function?

• Ellipsis

• U

• Parabola

• Asymptote

All of these forms still create a parabola!

f(x) = x² - 2x - 3

f(x) = (x - 1)² - 4

f(x) = (x + 1)(x - 3)

Axis of Symmetry

The Axis of Symmetry is a vertical line that is found at the x-coordinate of the parabola's vertex.

Each pair of points on either side of the parabola are the same distance as each other from the axis of symmetry!

Fill in the Blank

If the vertex of a parabola is (4, -3), what is the axis of symmetry?

(Type your answer as "x = _" where the blank represents the value.)

=

Multiple Choice

Identify the axis of symmetry from the following quadratic function in vertex form:

f(x) = 4(x - 3)² + 6

• x = 4

• x = 3

• x = -3

• x = 6

X-Intercepts of a function

​Also sometimes called "roots," "zeros," or "solutions."

The x- intercepts are the "roots" of the function represented in factored form:
f(x) = a(x - r₁)(x - r₂)
These occur when the line of the graph crosses the x-axis.
Roots are found from the factors of the standard form, then x is solved for using the zero product principle.

Multiple Choice

Identify the roots of the following quadratic function in factored form:

f(x) = (x + 2)(x - 1)

• x = 2

  x = 1

• x = -2

  x = 1

• x = 2

  x = -1

• x = -2

  x = -1

Find the factors of the equation by reversing one of the FOIL or box methods.

​Standard form to Vertex form

We can move between forms
(but we always have to go through standard form to get there).

​Standard form to Factored form

​Factored form to Standard form

​Vertex form to Standard form

Complete the square to form the (x - h)² term, then account for the difference in the remaining coefficient.

Multiply the factors of the equation by using one of the FOIL or box methods.

Multiply the squared factor together using the FOIL or box method, then combine like terms for the remaining coefficient

​​​If a < 0 (negative),
concave down

​If a > 0 (positive),
concave up

Concavity depends on the sign of the leading coefficient.

Concave down,
maximum

​Concave up,
minimum

If the graph opens upwards, it has a minimum.

If it opens downwards, it has a maximum.

Multiple Choice

Identify whether the following quadratic function has a maximum or minimum:

f(x) = -x² - x - 1

• Maximum

• Minimum

Multiple Choice

Identify whether the following quadratic function has a maximum or minimum:

f(x) = (x + 2)² - 1

• Maximum

• Minimum

Multiple Choice

Identify whether the following quadratic function has a maximum or minimum:

f(x) = -2(x - 4)(x + 7)

• Maximum

• Minimum

Match

Match the forms of the quadratic function to their names:

• y = (x + 4)(x + 2)

• y = (x + 3)² - 1

• y = x² + 6x + 8

Factored form

Vertex form

Standard form

• Factored form

• Vertex form

• Standard form