Key Features of Quadratic Functions, Vertex Form, and Standard Form Honors Algebra 2

In this unit, we learn how to graph quadratic functions in different forms and how to solve quadratic equations and inequalities. Quadratic functions have three forms that we will consider:

Standard form

Vertex form

Factored form

What is common among all of these forms?

Recall the following about the quadratic parent function:

𝑓(𝑥) = 𝑥& Domain: Axis of symmetry: 𝑥 = 0 Vertex: (0,0) Range:

y-intercept: (0,0) x-intercept(s): (0,0) vertex is a minimum.

The maximum/minimum for a quadratic function that has an unrestricted domain occurs at the vertex of the parabola.

We have dealt with quadratic functions in the context of transformations:

while we didn’t put a name to the form, it is called vertex form.

This form is called vertex form because it is easy to read off the vertex – the vertex is .

y = ax2 +bx + c

y = a x −h( )2 +k

y = a x − p( ) x −q( )

(−∞,∞)

0,∞⎡⎣ )

y = a x −h( )2 +k

h,k( )

6

5

4

3

2

1

–4 –2 2 4

x y -3 9 -2 4 -1 1 0 0 1 1 2 4 3 9

What is the vertex? The vertex of a parabola is the point at which the quadratic function is either at a maximum/minimum AND the point about which the function is symmetric across a vertical line. The vertical line about which the quadratic function is symmetric (mirror image of one side of the function to the other) is called the axis of symmetry. Remember that an equation of the form 𝑥 = # gives us a vertical line at that x-value. As we should see in the graph below, if we find the axis of symmetry of a quadratic function, we can plot a point and its image (opposite the axis of symmetry) in order to obtain a part of the graph. As we plot more points, we get a clearer shape of the graph. The vertex of a quadratic function will always lie on the axis of symmetry. As such, in order to find the coordinates of the vertex of a quadratic function we use the x-value of the axis of symmetry, plug it into the function to find the y-value and the ordered pair is the vertex.

How does the leading coefficient (a) affect the graph of a quadratic function?

Example 1: Describe the transformations from the parent function (in the correct order) for each of the quadratic functions and determine any of the requested key features.

1.

Transformations: y-intercept: axis of symmetry: vertex: Is the vertex a maximum or minimum?

2.

Transformations: y-intercept: axis of symmetry: vertex: Is the vertex a maximum or minimum?

x,y( )

g(x)= −12

x +2( )2 h(x)=2 x −1( )2 +3

Example 2: What is the equation of a quadratic function with vertex (−2, 3) and y-intercept −1? Example 3: The height of a thrown ball is a quadratic function of the time it has been in the air. The graph of the quadratic function is the parabolic path of the ball. The vertex of the graph is and the path of the ball includes the point

. Write the equation of the quadratic function that models the trajectory of the ball.

Example 4: Graph each of the following quadratic functions and determine the indicated key features of the function. Describe the transformations of the function from the parent function (in the correct order).

1.

Transformations: axis of symmetry: vertex: Range: y-intercept: x-intercept(s):

1,20( )0,4( )

f (x)= 12

x −3( )2 −2

Quadratic Functions in Standard Form

standard form:

In order to graph the quadratic function given the equation in standard form, we need to determine a few key features of the function:

(1) the equation of the axis of symmetry (2) the coordinates of the vertex (3) a few additional points on the function.

The Axis of Symmetry

Recalling that the vertex of the parabola always lies on the axis of symmetry, we can find the y-value of the vertex by substituting the x-value of the vertex (axis of symmetry) into the equation for the quadratic function. Example 5: For each of the quadratic functions, determine the equation of the axis of symmetry and the coordinates of the vertex.

1.

Axis of Symmetry: x = __________

Vertex: ( ________ , ________ )

2. Axis of symmetry: Vertex:

Example 6: Find the equation of the parabola that passes through the points , , and .

f (x)= ax2 +bx + c

y = x2 + 4x +5 f (x)= 3x2 + 4x +6

−2,32( ) 1,5( ) 3,17( )

For a quadratic function in standard for , the axis of symmetry is given by .

y = ax2 +bx + c x = − b2a

For a quadratic function in standard form , the coordinates of the vertex are

*− 𝒃𝟐𝒂, 𝒇 0− 𝒃

𝟐𝒂12.

y = ax2 +bx + c − b2a

, f − b2a

⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟

Example 7: Graph the indicated quadratic function and determine the indicated key features.

1. axis of symmetry: vertex: Range: y-intercept: x-intercept(s):

2. axis of symmetry: vertex: Range: y-intercept: x-intercept(s):

y = x2 − 4x +8

y = −2x2 +6x +8

The Connection between Vertex Form and Standard Form Quadratic functions have equations that can be represented in many forms – two of which we have studied so far are: vertex form standard form

We should be able to convert between vertex form and standard form – how? Example 4:

1. If , then write the quadratic function in standard form.

2. If , then write the quadratic function in vertex form.

y = a x −h( )2 +k y = ax2 +bx + c

p(x)= −3 x − 4( )2 −9

y = −2x2 +6x +8