Characteristics of Quadratics

Characteristics of Quadratics

By Mr. Jones

The direction of a quadratic describes how it opens: up or down ​

​

"A smile or a frown"​

Multiple Choice

What is the direction of the parabola?

• Opens Up

• Opens Down

• Opens Left

• Opens Right

Multiple Choice

What is the direction of the parabola?

$$3x^2-7x+8$$  

• Opens Up

• Opens Down

• Opens Left

• Opens Right

The "turning point"​

• The vertex of ​a quadratic is the point where the quadratic changes direction

Minimum or Maximum​

• The vertex can also be thought as the highest or lowest number.

• The minimum or maximum of a parabola is a number. Specifically, It is the y value of the vertex.

Multiple Select

Identify the vertex and then tell whether it's a minimum or maximum. Select two answer choices.

• (0,0)

• (1,0)

• (0.5,-1)

• Minimum

• Maximum

Multiple Select

Identify the vertex and then tell whether it's a minimum or maximum. Select two answer choices.

$$-x^2+10x-24$$

• (-4,0)

• (5,1)

• (6,0)

• Minimum

• Maximum

Multiple Choice

What is the maximum value of the quadratic?

• 3

• 0

• 4

• 1

• 5

Fill in the Blank

The table represents a quadratic function. Which point is the vertex?

(Remember vertex is the "turning point")

*Type a coordinate point

"Cuts the parabola in half"​

• The axis of symmetry is a vertical line that cuts the parabola in half

• The axis of symmetry is always in the form (x = a number)

• The number is the x-value of the vertex

Fill in the Blank

What is the axis of symmetry for the quadratic function shown?

*Type a line

Fill in the Blank

What is the axis of symmetry for the quadratic function shown?

*Type a line

"Touches the x-axis"​

• The x-intercept of a parabola is the point where the parabola intersects the x-axis

• A parabola can have one, two or zero x-intercepts

​Notice the y value of the x-intercept is always 0.

Solutions, Zeros, and Roots

• Once we have the x-intercept, we can identify the solutions of the quadratics. They are just the x-value of the x-intercept(s).

• Solutions, zeroes, and roots all mean the same thing!

Multiple Select

Identify the solutions of the quadratic shown

*May be more than one answer

• 5

• 1

• 0

• 3

• -4

Multiple Select

Identify the zeroes of the following quadratic

$$\left(x-3\right)^2$$  

*May be more than one answer.

• 0

• 3

• 9

• There are none.

Multiple Choice

How many roots can be identified from the quadratic table shown?

 

• None

• One

• Two

• Three

"Touches the y-axis"​

• The y-intercept of a parabola is the point where the parabola intersects the y-axis

• A parabola will always have one y-intercept.

​Notice how the x value of the y-intercept is always 0.

Multiple Choice

Which point represents the y-intercept of the quadratic?

• (-1,0)

• (-2,0)

• (-1.5,-0.25)

• (0,2)

Multiple Choice

Which point represents the y-intercept of the quadratic?

$$-x^2-7x-12$$  

• (-3.5, 0.25)

• (-4,0)

• (-3,0)

• (0,-12)

Multiple Select

The table shown is a quadratic function. Identify the y-intercept.

*May be more than one answer

 

• (-2,0)

• (-1,-3)

• (0,-4)

• (1,-3)

• (2,0)

Domain

• Remember the domain is all the possible x-values of a function.

• So we need to look at the x-axis

  • LEFT to RIGHT

• For parabolas, the domain will typically be all real numbers

Range​

• Remember the range is all the possible y-values of a function.

• So we need to look at the y-axis

  • UP and DOWN

• For parabolas, the range will never be all real numbers

• ​The min or max of a parabola can help you determine the range

Multiple Choice

What is the domain and range of the parabola?

*There are understood arrows on the ends

• Domain: $$\left(-\infty,\infty\right)$$  

  Range: $$y\ge3$$  

• Domain: $$x\ge3$$  

  Range: $$\left(-\infty,\infty\right)$$  

• Domain: $$\left(-\infty,\infty\right)$$  

  Range: $$y\le3$$  

• Domain: $$\left(-\infty,\infty\right)$$  

  Range: $$\left(-\infty,\infty\right)$$  

• Domain: $$x\le3$$  

  Range: $$\left(-\infty,\infty\right)$$  

Multiple Choice

What is the domain and range of the parabola?

*There are understood arrows on the ends

• Domain: $$\left(-\infty,\infty\right)$$  

  Range: $$y\ge5$$  

• Domain: $$x\ge5$$  

  Range: $$\left(-\infty,\infty\right)$$  

• Domain: $$\left(-\infty,\infty\right)$$  

  Range: $$y\le5$$  

• Domain: $$\left(-\infty,\infty\right)$$  

  Range: $$\left(-\infty,\infty\right)$$  

• Domain: $$x\le5$$  

  Range: $$\left(-\infty,\infty\right)$$