QUADRATIC FUNCTIONS

What is the name of the

quadratic function shown in

the graph?

Can you identify its

attributes?

QUADRATIC FUNCTIONS

DOMAIN – The set of possible input values.

RANGE – The set of possible output values.

2 TYPES OF NOTATION INTERVAL INEQUALITY

DOMAIN

RANGE

The height of a ball thrown up from a platform is shown versus time on the graph. What is the domain and range of this function? Domain Range

The graph of a quadratic function is shown to the right. Complete the list of attributes:

Domain

Range

X-intercepts

Vertex

Y-intercept

Maximum/Minimum

Line of Symmetry

The graph shows the path of a punted football. What is the maximum height that the football reaches? What is the reasonable domain? What is the reasonable range?

Write the domain and range of the quadratic equation, y = -x2 , in inequality notation. Domain Range

What are the domain and range for the quadratic equations shown below?

Equation Interval Notation Inequality Notation

Domain Range Domain Range

y = x2

y = x2 - 4

y = -2x2

y = -3x2 + 2

What are the domain and range of the quadratic function shown in the graph? Domain Range

EFFECTS OF CHANGES ax2 + bx + c

CONSTANT COEFFICIENT (c)

The value of c determines the y-value of the y-intercept.

For the graph of y = ax2 + c, the value of c determines the y-

coordinate of the vertex of the parabola.

If c ≠ 0, then the value of c determines how far up or down

(translates or shifts) along the y-axis as it compares to the parent

function, y = x2. When the parabola translates, the shape does not

change.

If c is positive, the graph shifts (translates) up.

If c is negative, the graph shifts (translates) down.

EFFECTS OF CHANGES ax2 + bx + c

QUADRATIC COEFFICIENT (a)

The value of a determines the width and direction of the parabola.

Width

As the absolute value of a increases (a > 1), the graph gets narrower

(constricts or condenses).

As the absolute value of a decreases (0 < a < 1, between 0 & 1), the

graph gets wider (expands).

Direction

If a is positive, the graph opens upward and can be described as

minimum.

Minimum – the lowest point on the graph (see the vertex) and the point

with the smallest y-value.

If a is negative (a < 1), the graph opens downward (reflects) and can be

described as maximum.

Maximum - the highest point on the graph (see the vertex) and the point

with the largest y-value.

How will the graph of y = x2 – 1 be affected if the quadratic coefficient is changed to 2?

How will the graph of y = 3x2 + 1 be affected if the quadratic coefficient is changed to 2? How will the graph of y = -2x2 + 2 be affected if the quadratic coefficient is changed to -3?

Rearrange the following equations in order from narrowest to widest; y = -2x2,

y = x2, y = 3x2, y = .5x2, and y = −𝟏

𝟑x2.

In the graph to the right, how does the graph y = -x2 - 1 compare to the graph of y = x2 - 1?

The graph of the parabola y = 3x2 + 3 is reflected over a line to obtain the graph of y = -3x2 + 3 . What is the line of reflection? What is the equation of a parabola that is a reflection across the x-axis of the parabola y = 4x2?

If the constant term of the graph y = x2 + 3 is changed to -1, describe the effect. How would the graph of y = x2 + 5 be changed if the function were changed to y = x2 + 10? In the graph of the function y = x 2 – 3, describe the shift in the vertex of the parabola if the -3 is changed to 8.

Hints: For y = x2 + c, always subtract the original value of c from the new value of c to find the vertical change. When the graph of a parabola is translated vertically by changing the value of c, the graph moves as a whole. Each point is translated the same distance as the other point.

DEMONSTRATION OF LEARNING A stone was dropped from a rooftop that is 16 feet above the ground. Its height, h, in feet after t seconds can be represented by the function h = -16t2 + 16.

For the equation, x2 – 10x – 24, identify the following parameters: X-intercepts Domain Y-intercept Range Vertex Minimum or Maximum Line of Symmetry Number of Solutions

Reasonable domain Reasonable range

Joe has a rectangular deck in his backyard. Its length measures 1 foot more than its width. He is planning to extend the length of the deck by 3 additional feet. If the new deck would have an area of 165 feet, what is the width of the deck.

The solutions of a quadratic equation are 3 and -2. Determine the factors and the standard form equation if a = 1.

The sum of the solutions of a quadratic equation is 𝟑

𝟐. The product of its

solutions is 3. What is the equation?

Kurt has a rectangular patio that is 12 feet long and 10 feet wide. He wants to increase the length and the width so the area of the new patio will be twice that of the original patio. The length will be increased by twice the amount that the width is increased. By how much should he increase each dimension?

Lucy just put a mat around her new Elvis poster. If the matted poster is 19-by-26 inches, and there is 408 ins2 of the poster showing, how wide is the mat?

Extra Graphs