10-1/10-2 Graphing Quadratic FunctionsBy Ms. Hall

Do you see a parabola? Can you see a non-function?

10-1 Exploring Quadratic Functions

The graph of a quadratic function is a parabola or u-shape.A parabola can open up or down.

If the parabola opens up, the lowest point is called the vertex. The vertex is a minimum.

If the parabola opens down , the vertex is the highest point. The vertex is a maximum.

y

x

NOTE: if the parabola opened left or right it would not be a function! It fails the vertical line test!

Vertex

Vertex

Line of SymmetryWhere have you heard this before?

y

x

Line of Symmetry

The line of symmetry ALWAYS passes through the vertex.

Parabolas have a symmetric property to them.

If we drew a line down the middle of the parabola, we could fold the parabola in half.

We call this line the line of symmetry.

Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.

Vertex

Example: Identify the vertex of each graph. Tell whether it is a minimum or

a maximum. Can you tell me how you are going to tell the difference?

a.

The vertex is (1, 2).

b.

The vertex is (2, –4).

It is a maximum. It is a minimum.

Let’s make some quadratic graphs!

Go to the next page and find the blank space at the bottom of the paper. You should see the heading Examples 1-5.

Let’s graph Example 1. ***Be sure and label it.

Example 1: Make a table of values and graph the quadratic

function y = x2. You pick values for x then plug them back into the

equation to find y. We are going to start by picking 3 values for x but

we may have to pick more until we see our parabola.

13

x y = x2 (x, y)13

0 (0)2 = 0 (0, 0)13

2 (2)2 = 1 (2, 1 )13

13

13

3 (3)2 = 3 (3, 3)13

Example 2: Make a table of values and graph the quadratic

function y = -2x2. You pick 3 values for x then plug them back into the

equation to find y. Again, you might need to pick more to make sure

you see your parabola!

x y = -2x2 (x, y)

0 -2(0)2 = 0 (0, 0)

1 -2(1)2 = -2 (1, -2)

2 -2(2)2 = -8 (2, -8)

Various Forms of Quadratic Equations

Vertex Form is one such form.What are the other forms…patience, patience!

Vertex form is awesome because it allows you to see the vertex which is the MIDDLE of the graph. So you instantly know your vertex as well as your line of symmetry!

So how would that help you graph?

Vertex Form

Vertex Form

(h, k) is the vertex.

So what does it have to do with quadratic graphs?

Let’s compare two quadratic functions and see if we notice anything.

Let’s start by comparing the most basic quadratic equation y = x2 and others.

Example 3: Graph the quadratic functions y = 3x2 and y = 3x2 – 2. Compare the

graphs.

The graph of y = 3x2 – 2 has the same shape as the graph of y = 3x2, but it is shifted down 2 units.

x y = 3x2 y = 3x2 – 2

2 12 10

1 3 1

0 0 2

-1 3 1

-2 12 10

Let’s start by comparing the most basic quadratic equation y = x2 and

others

Example 4: Graph the quadratic functions y = 4x2 and y = 4x2 + 3. Compare the

graphs.

The graph of y = 4x2 + 3 has the same shape as the graph of y = 4x2, but it is shifted up 3 units.

x y = 4x2 y = 4x2 + 3

2 16 19

1 4 7

0 0 3

-1 4 7

-2 16 19

Example: (x) = –x2, (x) = –3x2, and (x) = x2 Which one is the

widest? Narrowest?

So, the order from widest to narrowest is (x) = x2, (x) = –x2,

(x) = –3x2.

12

12

(x) = –x2(x) = x2 1

2

Of the three graphs, (x) = x2 is the widest and (x) = –3x2 is the narrowest.

12

(x) = –3x2

y = a(x-h)2 + k

a tells you how wide or narrow the parabola is.

If |a| < 1, then the parabola is wide.

If |a| > 1, then the parabola is narrow. y

x

a tells you if the parabola opens up or down.

If a is positive, then the parabola opens up.

y

x

y

x

𝒚=𝒙𝟐

𝒚=−𝒙𝟐

If a is negative, then the parabola opens down.

Ex. y =

Ex. y =

What about h and k? y = a(x-h)2 + k

k moves the parabola up and down the y-axis.

To move the parabola up, k needs to be positive.

To move the parabola down, k needs to be negative.

Ex.

y

xEx.

h moves the parabola left and right across the x-axis.

To move the parabola to the left, h needs to be positive.

To move the parabola to the right, h needs to be negative.

Ex.

Ex.

y

x

Height h is dependent on time t.

Graph t on the x-axis and h on the y-axis.

Use positive values for t.

Example 5: A monkey drops an orange from a branch 26 ft above the ground. The

force of gravity causes the orange to fall toward Earth. The function h = –16t2 + 26

gives the height of the orange, h, in feet after t seconds. Graph this quadratic

function.

t h = –16t2 + 260 261 102 –38