Graph Quadratic Functions and Equations

Finding the Parts of a Parabola

Learning Objectives:

At the end of the lesson, students will be able to:

1) Identify the parts of parabola
2) Solve for the vertex of a parabola
3) Graph a parabola
4) Apply parabola in real-life situation

Icebreaker Question:

​Maximum Profit

The profit on your school fundraiser is represented by the quadratic expression -3p² + 200p - 3000, where p is your price point. What price point will result in the maximum profit and what is the maximum profit?

Parabola: the graph of quadratic equation.
x-intercepts: the solution of quadratic equation.
Vertex: the maximum or minimum value.
Axis of symmetry: the vertical line that passes through the vertex.

​Parts of a Parabola

y-intercept: is a point where the parabola passes through the y-axis. (x = 0)
Remember:
If you are given the x-intercepts and the vertex, you can always graph the a parabola.

​Parts of a Parabola

Finding the Vertex of a Parabola

Plug in the value of x-coordinate to the equation to find y-coordinate.
y = (1)² - 2(1) - 3
= -4

Vertex: (1, -4)

Determine the values of a, b, and c.

Example:
y = x² - 2x - 3
a = 1, b = -2, c = -3

Step 1:

Step 2:

Step 3:

Find x-coordinate using x = -b/2a.

x = -b/2a
= -(-2)/2(1)
= 1

Solving for x-intercepts (zeroes, solution)

Write them as an ordered pair.

the x-intercepts:
(3, 0) and (-1, 0)

Set y = 0

Example:
y = x² - 2x - 3
0 = x² - 2x - 3

Step 1:

Step 2:

Step 3:

Solve for x.
(by factoring, completing the square, or quadratic formula)
x² - 2x - 3 = 0
(x - 3)(x + 1) = 0
x - 3 = 0 or x + 1 = 0
x = 3 or x = -1

Solving for y-intercepts (x = 0)

Set x = 0

Example:
y = x² - 2x - 3
y = 0² - 2(0) - 3
y = -3

Step 1:

Step 2:

Write then as an ordered pair.

y-intercept:
(0, -3)

Step 1:
Plot the vertex and two x-intercepts.

vertex:
(1, -4)
x-intercepts:
(3,0) and (-1, 0)

​

Graphing

Step 2:
Plot the y-intercepts.

​

Graphing

Step 3:
Connect the points to form the parabola.

​

Graphing

Icebreaker Question

To find the vertex, use x = -b/2a
x = -(200)/2(-3)
= -200/-6
= 33.33
Plug this into the equation.

Earlier you were asked to find the maximum point that will result in the maximum profit and to find the maximum profit.

The maximum profit occurs at the maximum point of the parabola, so find the vertex of -3p² + 200p - 3000.

Icebreaker Question

The maximum profit occurs at a price point of $33.33. At that price point, the profit would be $334.

Earlier you were asked to find the maximum point that will result in the maximum profit and to find the maximum profit.

The maximum profit occurs at the maximum point of the parabola, so find the vertex of -3p² + 200p - 3000.

To find the vertex, use x = 33.33
y = (-3)(33.33)² + (200)(33.33) - 3000
= -3333 + 6667 - 3000
= 334.
Therefore the vertex is (33.33, 334)

Real World Application:
You throw your ball into the air from a height of 4.2 feet with an initial vertical velocity of 24 feet per second. Use the vertical model, h = -16t² + vt + s, where v is the initial velocity in feet per second and s is the height in feet, to calculate the maximum height of the ball.

Maximum Height: ____feet.

Real World Application:
You throw your ball into the air from a height of 4.2 feet with an initial vertical velocity of 24 feet per second. Use the vertical model, h = -16t² + vt + s, where v is the initial velocity in feet per second and s is the height in feet, to calculate the maximum height of the ball.

Maximum Height: 3.2 feet.

Real World Application:
Tyler kicks a football into the air from a height of 3 feet with an initial velocity of 48 feet per second. Use the vertical motion model, h = -16t² + vt + s, where v is the initial velocity in feet per second and s is the height in feet, to calculate the maximum height of the football.

Maximum Height: ______ feet.

Real World Application:
Tyler kicks a football into the air from a height of 3 feet with an initial velocity of 48 feet per second. Use the vertical motion model, h = -16t² + vt + s, where v is the initial velocity in feet per second and s is the height in feet, to calculate the maximum height of the football.

Maximum Height: 39 feet.