Chapter 10Quadratic Equations

& FunctionsExamine Graphs & Their Equations

Solve Quadratic Equations by Graphing, Factoring, & Using the

Quadratic Formula

Section 10 – 1 Exploring Quadratic

EquationsObjectives:

To graph quadratic functions of the form

Standard Form of a Quadratic Function

𝒚=𝒂𝒙𝟐+𝒃𝒙+𝒄Examples:

Parabola:The U-Shaped Curve the graph of a quadratic function makes.

Axis of Symmetry:The line (sometimes invisible) that divides the parabola into two matching halves.

Vertex:The highest or lowest point of the parabola, found on the axis of symmetry. (The bottom or top of the U).

Minimum Vs. MaximumWhen the Parabola Opens UP (like a U),

the vertex is at the bottom. We call this a minimum.

When the Parabola Opens DOWN (like an A), the vertex is at the top. We call

this a maximum.

Minimum Vs. Maximum(U) Minimum: When a > 0 (Positive)

(A) Maximum: When a < 0 (Negative)

Example 1 Identifying a VertexIdentify the vertex of each graph. Tell whether it is a minimum or maximum.

A) B)

C) D)

Example 2 Graphing Make a table of values and graph each quadratic function.A)

X (x, y)

0

2

4

Functions in the form have a vertex at (0, 0)!Plot a few points on one side of the vertex, and then reflect each point across the axis

of symmetry!

B)

X (x, y)

0

1

2

Functions in the form have a vertex at (0, 0)!Plot a few points on one side of the vertex, and then reflect each point across the axis

of symmetry!

C)

X (x, y)

0

2

3

Functions in the form have a vertex at (0, 0)!Plot a few points on one side of the vertex, and then reflect each point across the axis

of symmetry!

Example 3 Comparing Widths of ParabolasA) Use the graphs below. Order the quadratic functions from widest to narrowest graph.

𝑦=−4 𝑥2 𝑦=14 𝑥

2𝑦=𝑥2

B) Use the graphs below. Order the quadratic functions from widest to narrowest graph.

C) Order the quadratic functions from widest to narrowest graph.

𝑦=𝑥2 𝑦=12 𝑥

2𝑦=−2𝑥2

D) Order the quadratic functions from widest to narrowest graph.

𝑦=−12 𝑥2 𝑦=

13 𝑥

2𝑦=−4 𝑥2

Homework

Textbook Page 513; #1 – 13 All

(Use Graph Paper for #4 – 9)

Warm Up

2) Order the quadratic functions from widest to narrowest graph.

𝑦=−12 𝑥2 𝑦=

13 𝑥

2𝑦=−4 𝑥2

1) Name the vertex of each parabola, then determine whether it is a minimum or

maximum.

Section 10 – 1 Continued…

Objectives:To graph quadratic functions

of the form

𝑦=𝑎𝑥 2+𝒄The value of C, the constant term in a quadratic function,

translates the graph up or down!

Example 4 Graphing Graph each quadratic function. Compare the Graphs.A) and

X

0

2

4

Graph each quadratic function. Compare the Graphs.B) and

X

0

2

4

Real- World ConnectionYou can model the height of an object moving

under the influence of gravity using a quadratic function. As an object falls, its speed continues to

increase. Ignoring air resistance, you can approximate height of a falling object using a

function . The height (h) is in feet, the time (t) is in seconds, and the initial height of the object (c)

is in feet.

Example 5 Graphing A) Suppose you see an eagle flying over a canyon. The eagle is 30 feet above the level of the canyon’s edge when it drops a stick from its claws. The force of gravity causes the stick to fall toward the Earth. The function gives the height of the stick (h) in feet after (t) seconds. Graph this quadratic function.

B) Suppose a squirrel is in a tree 24 feet above the ground. She drops an acorn. The function give the height of the acorn in feet after t seconds. Graph this function.

Homework

• 10 – 1 Ditto; 2 – 30 Even