wow! nothing made sense! yay! :D SAT is in one less than month AND I DONT UNDERSTAND PARABOLAS! T^T

august 26th DSAT and i dont understand transformations

i didn't understand ANYTHING!!

Try the quadratics unit in Khan Maths.

Main content

Course: Digital SAT Math > Unit 12

Lesson 11: Quadratic graphs: advanced

Quadratic graphs | Lesson

Google Classroom

A guide to quadratic graphs on the digital SAT

What are quadratic functions?

In a quadratic function, the

of the function is based on an expression in which the

is the highest power term. For example,

f (x) = x^{2} + 2 x + 1

is a quadratic function, because in the highest power term, the

x

is raised to the second power.

Unlike the graphs of linear functions, the graphs of quadratic functions are nonlinear: they don't look like straight lines. Specifically, the graphs of quadratic functions are called parabolas.

In this lesson, we'll learn to:

Graph quadratic functions
Identify the features of quadratic functions
Rewrite quadratic functions to showcase specific features of graphs
Transform quadratic functions

You can learn anything. Let's do this!

How do I graph parabolas, and what are their features?

Parabolas intro

Khan Academy video wrapper

Parabolas introSee video transcript

What are the features of a parabola?

All parabolas have a

y

-intercept, a

, and open either upward or downward.

Since the vertex is the point at which a parabola changes from increasing to decreasing or vice versa, it is also either the maximum or minimum $y$ ‍-value of the parabola.

If the parabola opens upward, then the vertex is the lowest point on the parabola.
If the parabola opens downward, then the vertex is the highest point on the parabola.

A parabola can also have zero, one, or two $x$ ‍-intercepts.

Note: the terms "zero" and "root" are used interchangeably with "

x

-intercept". They all mean the same thing!

Parabolas also have vertical symmetry along a vertical line that passes through the vertex.

For example, if a parabola has a vertex at

(2, 0)

, then the parabola has the same

y

-values at

x = 1

and

x = 3

, at

x = 0

and

x = 4

, and so on.

To graph a quadratic function:

Evaluate the function at several different values of $x$ ‍.
Plot the input-output pairs as points in the $x y$ ‍-plane.
Sketch a parabola that passes through the points.

Example: Graph

f (x) = x^{2} - 3

in the

x y

-plane.

First, let's evaluate the function at several different values of

x

$x$ ‍	$f (x)$ ‍
$- 1$ ‍	$(- 1)^{2} - 3 = - 2$ ‍
$0$ ‍	$(0)^{2} - 3 = - 3$ ‍
$1$ ‍	$(1)^{2} - 3 = - 2$ ‍
$2$ ‍	$(2)^{2} - 3 = 1$ ‍

The graph contains the points

(- 1, - 2)

(0, - 3)

(1, - 2)

, and

(2, 1)

Based on these four points, as

x

increases,

y

decreases, then increases. We can sketch a parabola that passes through the four points:

We'll learn more about finding the exact coordinates of the parabola's features in the next section. Sometimes, simply having a rough sketch of the graph can come in handy!

Try it!

TRY: match the features of the parabola to their coordinates

The graph of

y = - x^{2} + 4 x + 5

is shown above. Match the features of the graph with their coordinates.

1

How do I identify features of parabolas from quadratic functions?

Forms & features of quadratic functions

Khan Academy video wrapper

Forms & features of quadratic functionsSee video transcript

Standard form, factored form, and vertex form: What forms do quadratic equations take?

For all three forms of quadratic equations, the coefficient of the

x^{2}

-term,

a

, tells us whether the parabola opens upward or downward:

If $a > 0$ ‍, then the parabola opens upward.
If $a < 0$ ‍, then the parabola opens downward.

The magnitude of

a

also describes how steep or shallow the parabola is. Parabolas with larger magnitudes of

a

are more steep and narrow compared to parabolas with smaller magnitudes of

a

, which tend to be more shallow and wide.

The graph below shows the graphs of

y = a x^{2}

for various values of

a

The standard form of a quadratic equation,

y = a x^{2} + b x + c

, shows the

y

-intercept of the parabola:

The $y$ ‍-intercept of the parabola is located at $(0, c)$ ‍.

The factored form of a quadratic equation,

y = a (x - b) (x - c)

, shows the

x

-intercept(s) of the parabola:

$x = b$ ‍ and $x = c$ ‍ are solutions to the equation $a (x - b) (x - c) = 0$ ‍.
The $x$ ‍-intercepts of the parabola are located at $(b, 0)$ ‍ and $(c, 0)$ ‍.
The terms $x$ ‍-intercept, zero, and root can be used interchangeably.

The vertex form of a quadratic equation,

y = a (x - h)^{2} + k

, reveals the vertex of the parabola.

The vertex of the parabola is located at $(h, k)$ ‍.

To identify the features of a parabola from a quadratic equation:

Remember which equation form displays the relevant features as constants or coefficients.
Rewrite the equation in a more helpful form if necessary.
Identify the constants or coefficients that correspond to the features of interest.

Example: What are the zeros of the graph of

f (x) = x^{2} + 7 x + 12

The factored form of a quadratic equation shows the zeros of the parabola. We can rewrite the function in factored form and identify the zero.

x^{2} + 7 x + 12

can be factored into

(x + a) (x + b)

such that:

$a + b = 7$ ‍
$a b = 12$ ‍

a = 3

and

b = 4

would work:

$3 + 4 = 7$ ‍
$(3) (4) = 12$ ‍

The zeros of the parabola correspond to the solutions of

0 = (x + 3) (x + 4)

\begin{aligned} 0 & = (x + 3) (x + 4) \\ 0 & = x + 3 or 0 = x + 4 \\ - 3 & = x or - 4 = x \end{aligned}

The zeroes of the graph are $- 3$ ‍ and $- 4$ ‍.

To match a parabola with its quadratic equation:

Determine the features of the parabola.
Identify the features shown in quadratic equation(s).
Select a quadratic equation with the same features as the parabola.
Plug in a point that is not a feature from Step 2 to calculate the coefficient of the $x^{2}$ ‍-term if necessary.

Example:

What is a possible equation for the parabola shown above?

Based on the graph, the parabola opens downward, has a vertex located at

(- 3, 3)

, and has zeros located at

(- 4, 0)

and

(- 2, 0)

We can write an equation of the parabola in vertex form,

y = a (x - h)^{2} + k

. Since the vertex is located at

(- 3, 3)

h = - 3

and

k = 3

\begin{aligned} y & = a (x - (- 3))^{2} + 3 \\ y & = a (x + 3)^{2} + 3 \end{aligned}

But we're not done yet! We still need to determine the value of

a

. We can do so by substituting the coordinates of either zero into the equation and solving for

a

. Using

(x, y) = (- 2, 0)

\begin{aligned} 0 & = a (- 2 + 3)^{2} + 3 \\ 0 & = a (1)^{2} + 3 \\ 0 & = a + 3 \\ 0 - 3 & = a + 3 - 3 \\ - 3 & = a \end{aligned}

The equation of the parabola in vertex form is:

y = - 3 (x + 3)^{2} + 3

We can also rewrite the equation in standard form or factored form:

\begin{aligned} y & = - 3 (x + 3)^{2} + 3 \\ = - 3 (x^{2} + 6 x + 9) + 3 \\ = - 3 x^{2} - 18 x - 27 + 3 \\ y & = - 3 x^{2} - 18 x - 24 & standard form \\ = - 3 (x^{2} + 6 x + 8) \\ y & = - 3 (x + 2) (x + 4) & factored form \end{aligned}

Notice that we can see the zeros of the parabola displayed as constants in the factored form of the equation.

$y = - 3 (x + 3)^{2} + 3$ ‍ is a possible equation for the parabola.

Try it!

TRY: determine the feature of a parabola from its equation

The graph of the equation

y = \frac{1}{2} x^{2} - 7

is a parabola that opens

because the coefficient of

x^{2}

TRY: determine the feature of a parabola from its equation

The

of the graph of

f (x) = (x + 2)^{2} + 7

is located at the point

(- 2, 7)

7

is the

y

-value of the graph.

TRY: determine the feature of a parabola from its equation

y = (x - 2) (x - 4)

is the

form of a quadratic equation. The

of the graph can be identified as constants in the equation.

How do I rewrite quadratic functions to reveal specific features of parabolas?

Equivalent forms of quadratic functions

When we're given a quadratic function, we can rewrite the function according to the features we want to display:

$y$ ‍-intercept: standard form
$x$ ‍-intercept(s): factored form
Vertex: vertex form

When rewriting a quadratic function to display specific graphical features:

Choose the appropriate quadratic form based on the graphic feature to be displayed.
Rewrite the given quadratic expression as an equivalent expression in the form identified in Step 1.

Example:

The graph of

y = x^{2} - 4 x - 5

is shown above. Write an equivalent equation from which the coordinates of the vertex can be identified as constants in the equation.

According to the graph, the vertex of the parabola is located at

(2, - 9)

. We can start by substituting

2

for

h

and

- 9

for

k

into the vertex form equation

y = a (x - h)^{2} + k

y = a (x - 2)^{2} - 9

Since the

x^{2}

-coefficient of

x^{2} - 4 x - 5

1

a

must also be equal to

1

. We can confirm that that

x^{2} - 4 x - 5

and

(x - 2)^{2} - 9

are equivalent:

\begin{aligned} x^{2} - 4 x - 5 & \overset{?}{=} (x - 2)^{2} - 9 \\ x^{2} - 4 x - 5 & \overset{?}{=} x^{2} - 4 x + 4 - 9 & square of difference \\ x^{2} - 4 x - 5 & \overset{✓}{=} x^{4} - 4 x - 5 \end{aligned}

The coordinates of the vertex can be identified as constants in the equation $y = (x - 2)^{2} - 9$ ‍.

Try it!

Try: identify the appropriate forms to use

A graph of the quadratic equation

y = x^{2} - 2 x - 3

is shown in the

x y

-plane above.

The vertex of the parabola is located at

. To show the coordinates of the vertex as constants or coefficients, we should use the

form of the quadratic equation,

y = (x - 1)^{2} - 4

The

x

-intercepts of the parabola are

. To show the

x

-intercepts as constants or coefficients, we should use the

form of the quadratic equation,

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

How do I transform graphs of quadratic functions?

Intro to parabola transformations

Khan Academy video wrapper

Intro to parabola transformationsSee video transcript

Translating, stretching, and reflecting: How does changing the function transform the parabola?

We can use function notation to represent the translation of a graph in the

x y

-plane. If the graph of

y = f (x)

is graphed in the

x y

-plane and

c

is a positive constant:

The graph of $y = f (x - c)$ ‍ is the graph of $f (x)$ ‍ shifted to the right by $c$ ‍ units.
The graph of $y = f (x + c)$ ‍ is the graph of $f (x)$ ‍ shifted to the left by $c$ ‍ units.
The graph of $y = f (x) + c$ ‍ is the graph of $f (x)$ ‍ shifted up by $c$ ‍ units.
The graph of $y = f (x) - c$ ‍ is the graph of $f (x)$ ‍ shifted down by $c$ ‍ units.

The graph below shows the graph of the quadratic function

f (x) = x^{2} - 3

alongside various translations:

The graph of $f (x - 4) = (x - 4)^{2} - 3$ ‍ translates the graph of $4$ ‍ units to the right.
The graph of $f (x + 6) = (x + 6)^{2} - 3$ ‍ translates the graph $6$ ‍ units to the left.
The graph of $f (x) + 5 = x^{2} + 2$ ‍ translates the graph $5$ ‍ units up.
The graph of $f (x) - 3 = x^{2} - 6$ ‍ translates the graph $3$ ‍ units down.

We can also represent stretching and reflecting graphs algebraically. If the graph of

y = f (x)

is graphed in the

x y

-plane and

c

is a positive constant:

The graph of $y = - f (x)$ ‍ is the graph of $f (x)$ ‍ reflected across the $x$ ‍-axis.
The graph of $y = f (- x)$ ‍ is the graph of $f (x)$ ‍ reflected across the $y$ ‍-axis.
The graph of $y = c \cdot f (x)$ ‍ is the graph of $f (x)$ ‍ stretched vertically by a factor of $c$ ‍.

The graph below shows the graph of the quadratic function

f (x) = x^{2} - 2 x - 2

alongside various transformations:

The graph of $- f (x) = - x^{2} + 2 x + 2$ ‍ is the graph of $f (x)$ ‍ reflected across the $x$ ‍-axis.
The graph of $f (- x) = x^{2} + 2 x - 2$ ‍ is the graph of $f (x)$ ‍ reflected across the $y$ ‍-axis.
The graph of $3 \cdot f (x) = 3 x^{2} - 6 x - 6$ ‍ is the graph of $f (x)$ ‍ stretched vertically by a factor of $3$ ‍.

Try it!

TRY: Shift a parabola

Compared to the graph of

y = x^{2}

, the graph of

y = (x + 4)^{2} + 3

is shifted

4

units

and

3

units

Try: reflect a parabola

The graph of

y = f (x)

is a parabola that opens upward and has a vertex located at

(2, 1)

The graph of

y = f (- x)

has a vertex located at

The graph of

y = - f (x)

is a parabola that opens

Things to remember

Forms of quadratic equations

Standard form: A parabola with the equation

y = a x^{2} + b x + c

has its

y

-intercept located at

(0, c)

Factored form: A parabola with the equation

y = a (x - b) (x - c)

has its

x

-intercept(s) located at

(b, 0)

and

(c, 0)

Vertex form: A parabola with the equation

y = a (x - h)^{2} + k

has its vertex located at

(h, k)

When we're given a quadratic function, we can rewrite the function according to the features we want to display:

$y$ ‍-intercept: standard form
$x$ ‍-intercept(s): factored form
Vertex: vertex form

Transformations

If the graph of

y = f (x)

is graphed in the

x y

-plane and

c

is a positive constant:

The graph of $y = f (x - c)$ ‍ is the graph of $f (x)$ ‍ shifted to the right by $c$ ‍ units.
The graph of $y = f (x + c)$ ‍ is the graph of $f (x)$ ‍ shifted to the left by $c$ ‍ units.
The graph of $y = f (x) + c$ ‍ is the graph of $f (x)$ ‍ shifted up by $c$ ‍ units.
The graph of $y = f (x) - c$ ‍ is the graph of $f (x)$ ‍ shifted down by $c$ ‍ units.
The graph of $y = - f (x)$ ‍ is the graph of $f (x)$ ‍ reflected across the $x$ ‍-axis.
The graph of $y = f (- x)$ ‍ is the graph of $f (x)$ ‍ reflected across the $y$ ‍-axis.
The graph of $y = c \cdot f (x)$ ‍ is the graph of $f (x)$ ‍ stretched vertically by a factor of $c$ ‍.

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Digital SAT Math

Course: Digital SAT Math > Unit 12

What are quadratic functions?

How do I graph parabolas, and what are their features?

Parabolas intro

What are the features of a parabola?

Try it!

How do I identify features of parabolas from quadratic functions?

Forms & features of quadratic functions

Standard form, factored form, and vertex form: What forms do quadratic equations take?

Try it!

How do I rewrite quadratic functions to reveal specific features of parabolas?

Equivalent forms of quadratic functions

Try it!

How do I transform graphs of quadratic functions?

Intro to parabola transformations

Translating, stretching, and reflecting: How does changing the function transform the parabola?

Try it!

Your turn!

Things to remember

Forms of quadratic equations

Transformations

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