Introduction to QuadraticsIntroduction to QuadraticsObjectives:Objectives:Define Quadratic Functions and Parent functionsDefine Quadratic Functions and Parent functionsExplore Parameter changes and their effects.Explore Parameter changes and their effects.

R. Portteus 2005-06T. Merrill 2006

What is a Quadratic Function?What is a Quadratic Function?

A A quadratic functionquadratic function is any function is any function whose graph is a parabola.whose graph is a parabola.

A A quadratic equationquadratic equation is any equation that is any equation that can be written in the form y = axcan be written in the form y = ax22 + bx + c. + bx + c. The constants a, b, and c are called the The constants a, b, and c are called the parametersparameters of the equation. These of the equation. These values tell us the shape and location of the values tell us the shape and location of the parabola.parabola.

Parent FunctionsParent Functions

The The parent functionparent function is the simplest is the simplest function of a certain type. It is called this function of a certain type. It is called this because all of functions in that group look because all of functions in that group look like that and only change location and like that and only change location and shape.shape.

The linear parent function is y = x.The linear parent function is y = x. All graphs are straight lines.All graphs are straight lines.

The quadratic parent function is y = xThe quadratic parent function is y = x22.. All graphs are parabolas.All graphs are parabolas.

Effects on ParametersEffects on Parameters

What happens to the graph of y = axWhat happens to the graph of y = ax22 when a is changed?when a is changed? If a > 0, the parabola opens upward.If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.If a < 0, the parabola opens downward.

ExamplesExamples

x

y

x

y

y = 2x2 opens upward y = -2x2 opens downward

Graph the following four parabolas Graph the following four parabolas on the graph at right and label eachon the graph at right and label each

A.A.

B.B.

C.C.

D.D.

2xy 22xy

22xy

2xy

x

y

Effect of –a In a quadratic formula , when a is multiplied by -1 the resulting graph is the same as the original graph _____________________________.

cbxaxy 2

Reflected over the x-axis

Effects on ParametersEffects on Parameters

If two quadratic functions of the form If two quadratic functions of the form y = axy = ax22 have different coefficients, then have different coefficients, then one graph will be wider than the other.one graph will be wider than the other.

Which of these functions produce the Which of these functions produce the widest graph?widest graph? y = 3xy = 3x22, y = -5x, y = -5x22, y = ¾ x, y = ¾ x22

AnswerAnswer

x

y

y = ¾ x2

y = -5x2

y = 3x2

y = ¾ x2 is wider, so the smaller a is, the wider the graph is.

Graph the following four parabolas Graph the following four parabolas on the graph at right and label eachon the graph at right and label each

A.A.

B.B.

C.C.

D.D.

22xy 24xy 22xy

2

2

1xy

x

y

Effect of |a| In a quadratic formula , when |a| increases, the resulting graph is ___________, when |a| decreases, the resulting graph is _____________.

cbxaxy 2

narrowerwider

Effects on ParametersEffects on Parameters

If two quadratic functions of the form If two quadratic functions of the form y = (x - h)y = (x - h)22 have different values for h then have different values for h then one graph will be a translation right or left one graph will be a translation right or left from the other graph.from the other graph.

Compare these three graphs:Compare these three graphs:

y = (x – 0)y = (x – 0)22 y = (x + 1)y = (x + 1)22 y = (x – 4)y = (x – 4)22

Parent function!

•What happens to the parent function What happens to the parent function when we “add 1”? when we “add 1”?

•What happens when we “subtract four”?What happens when we “subtract four”?

x

y

AnswerAnswer

y = (x+1)2 y = x2 - 4

y = (x + 1)2 is translated left one spaces from y = x2

y = (x – 4)2 is translated right four spaces from y = x2

Effects on ParametersEffects on Parameters

If two quadratic functions of the form If two quadratic functions of the form y = xy = x22 + c have different constants, c, then + c have different constants, c, then one graph will be a translation up or down one graph will be a translation up or down from the other graph.from the other graph.

Compare the graphs of y = xCompare the graphs of y = x22 + 3 with the + 3 with the graph of y = xgraph of y = x22 – 4. – 4.

AnswerAnswer

x

y

y = x2 + 3

y = x2 - 4

y = x2 + 3 is translated up three spaces from y = x2 and y = x2 – 4 is translated down four spaces from y = x2, so there are 7 spaces in between the two graphs.

Graph the following four parabolas Graph the following four parabolas on the graph at right and label eachon the graph at right and label each

A.A.

B.B.

C.C.

D.D.

22 xxy

x

y

Effect of c In a quadratic formula , when c increases one unit, the resulting graph is __________________, when c decreases one unit, the resulting graph is __________________.

cbxaxy 2

52 xxy

22 xxy

52 xxy

translated up 1 unit translated down 1 unit

ExamplesExamples

Identify the parent function of the Identify the parent function of the following equations.following equations.

1.1. y = -3xy = -3x22 + 4x – 7? + 4x – 7?

y = xy = x22 →→ Quadratic Function Quadratic Function

2.2. 3x – 4y = 83x – 4y = 8

y = x y = x →→ Linear Function Linear Function

Examples - ContinuedExamples - Continued

Determine whether the function face upward Determine whether the function face upward or downward.or downward.

1.1. y = -3xy = -3x22 + 2 + 2

- Downward- Downward

2.2. y = ½ xy = ½ x22

- Upward- Upward

Examples- ContinuedExamples- Continued

What is the new equation if the given What is the new equation if the given function is translated down 4 spaces?function is translated down 4 spaces?

1.1. y = xy = x22 – 3 – 3

y = xy = x22 – 7 – 7

2.2. y = xy = x22 + 8 + 8

y = xy = x22 + 4 + 4

Examples - ContinuedExamples - Continued

Order the equations from widest to most Order the equations from widest to most narrow.narrow.

1.1. y = -6xy = -6x22, y = ¼ x, y = ¼ x22, y = 4x, y = 4x22

How do the two given equation compare?How do the two given equation compare?

2.2. y = -3xy = -3x22 – 8 y = x – 8 y = x22

y = ¼ xy = ¼ x22, y = 4x, y = 4x22, y = -6x, y = -6x22

Translated down 8 spaces, reflected over the x-axis Translated down 8 spaces, reflected over the x-axis to face downwards and is narrower.to face downwards and is narrower.

Lesson CheckLesson Check What causes a parabola to move left or right?What causes a parabola to move left or right?

What causes a parabola to move up or down?What causes a parabola to move up or down?

What causes a parabola to flip upside down?What causes a parabola to flip upside down?

What causes a parabola to get wider?What causes a parabola to get wider?

What causes a parabola to get narrower?What causes a parabola to get narrower?