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Parabolas & Quadratic Equations: Vertices, Graphing, and Solving Max/Min Problems, Study notes of Calculus for Engineers

University of Rio Grande Calculus for Engineers

Instructions on how to find the vertex of a parabola, graph a quadratic function, use the discriminant to determine the number of x-intercepts, and solve problems involving maximum or minimum values. It also covers parabolas with horizontal axes and completing the square. Written by Cindy Alder.

What you will learn

How do you find the vertex of a parabola?
How can you determine the number of x-intercepts of a quadratic function using the discriminant?
What is the difference between completing the square when a = 1 and when a ≠ 1?

How do you graph a quadratic function?
What is the formula for the vertex of a quadratic function?

Typology: Study notes

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9.6 More About

Parabolas & Their Applications

OBJECTIVES:

•Find the vertex of the vertical parabola.

•Graph a quadratic function.

•Use the discriminant to find the number of x-

intercepts of a parabola with a vertical axis.

•Use quadratic functions to solve problems

involving maximum or minimum values.

•Graph parabolas with horizontal axes.

Written by: Cindy Alder

Partial preview of the text

Download Parabolas & Quadratic Equations: Vertices, Graphing, and Solving Max/Min Problems and more Study notes Calculus for Engineers in PDF only on Docsity!

9.6 More About

Parabolas & Their Applications

OBJECTIVES:

Find the vertex of the vertical parabola.
Graph a quadratic function.
Use the discriminant to find the number of x-

intercepts of a parabola with a vertical axis.

Use quadratic functions to solve problems

involving maximum or minimum values.

Graph parabolas with horizontal axes.

Written by: Cindy Alder

Quadratic Equations

 We can tell a lot about a quadratic equation when it is in the form:

𝒇 𝒙 = 𝒂 𝒙 − 𝒉 𝟐^ + 𝒌

Vertex: Axis of Symmetry: Direction of opening: UP if DOWN if Wide/Narrow: WIDE if NARROW if Vertical Shift: Horizontal Shift:

axis of symmetry x = h

Vertex ( h, k )

Complete the Square when 𝒂 = 𝟏

𝒇 𝒙 = 𝒙𝟐^ − 𝟒𝒙 + 𝟓

Group the variable terms. Slide the constant over to “deal with later”.
Find the “magic” number needed to complete the square.
Add and subtract the “magic” number to the equation.
Factor and combine like terms.

Find the Vertex of the Graph of

𝒇 𝒙 = 𝒙𝟐^ + 𝟔𝒙 + 𝟏𝟒

Complete the Square when 𝒂 ≠ 𝟏

𝒇 𝒙 = −𝟑𝒙𝟐^ + 𝟔𝒙 − 𝟏

Group the variable terms. Slide the constant over to “deal with later”.
Factor out the coefficient on 𝒙𝟐^ from the variable terms.
Find the “magic” number needed to complete the square.
Add the “magic” number to the equation and
Subtract the product of the “magic” number and the coefficient you factored out from the equation.
Factor and combine like terms.

Find the Vertex of the Graph of

𝒇 𝒙 = 𝟒𝒙𝟐^ − 𝟑𝟐𝒙 + 𝟓𝟖

The Vertex Formula

𝒇 𝒙 = 𝒂𝒙𝟐^ + 𝒃𝒙 + 𝒄

Vertex Formula

−𝒃 𝟐𝒂

−𝒃 𝟐𝒂 𝒚 = 𝒇^ 𝒕𝒉𝒆 𝒗𝒂𝒍𝒖𝒆 𝒚𝒐𝒖 𝒈𝒐𝒕 𝒇𝒐𝒓 𝒙

Find the vertex using the vertex formula.

 𝒇 𝒙 = 𝒙𝟐^ + 𝟖𝒙 + 𝟏𝟎

Graphing a Quadratic Function

 STEPS  Does the graph open up or down?  Find the vertex.  Vertex formula  Complete the square  Find any intercepts.  y- intercept: set 𝑥 = 0  x- intercept: set 𝑦 = 0  Complete the graph.  Plot points found.  Plot additional points as needed, using symmetry.

_A minimum of two points on either side of the vertex is needed for an accurate graph._*

Graph the Quadratic Equation:

𝒇 𝒙 = 𝒙𝟐^ − 𝟔𝒙 + 𝟓

 Step 1 - Does the graph open up or down?

The graph will open up because a is positive.  Step 2 - Find the vertex.

+ +

 Step 4 - Complete the graph.  Plot points found.  Vertex: 3, −  y- intercept: 0,  x- intercepts: 5,0 𝑎𝑛𝑑 (1,0)

 Plot additional points as needed, using symmetry.  Using symmetry the graph will also pass through the point 6,.

Graph the Quadratic Equation (continued)

𝒇 𝒙 = 𝒙𝟐^ − 𝟔𝒙 + 𝟓

Graph the Function 𝒇 𝒙 = 𝒙𝟐^ − 𝒙 − 𝟔

Using the Discriminant

Find the discriminant and use it to determine the number of

x -intercepts of the graph of each quadratic function.

A) 𝑓 𝑥 = −3𝑥^2 − 𝑥 + 2

C) 𝑓 𝑥 = 𝑥^2 − 8𝑥 + 16

B) 𝑓 𝑥 = 𝑥^2 − 𝑥 + 1

Parabola with Horizontal Axis

The graph of

𝒙 = 𝒂𝒚𝟐^ + 𝒃𝒚 + 𝒄 or 𝒙 = 𝒂 𝒚 − 𝒌 𝟐^ + 𝒉

is a parabola with vertex (ℎ, 𝑘) and the horizontal line 𝑦 = 𝑘 as axis of symmetry. The graph opens to the right if 𝑎 > 0 and to the left if 𝑎 < 0.

Parabolas & Quadratic Equations: Vertices, Graphing, and Solving Max/Min Problems, Study notes of Calculus for Engineers

Related documents

Partial preview of the text

Download Parabolas & Quadratic Equations: Vertices, Graphing, and Solving Max/Min Problems and more Study notes Calculus for Engineers in PDF only on Docsity!

9.6 More About

Parabolas & Their Applications

OBJECTIVES:

intercepts of a parabola with a vertical axis.

involving maximum or minimum values.

𝒇 𝒙 = 𝒂 𝒙 − 𝒉 𝟐^ + 𝒌

𝒇 𝒙 = −𝟑𝒙𝟐^ + 𝟔𝒙 − 𝟏

A) 𝑓 𝑥 = −3𝑥^2 − 𝑥 + 2

C) 𝑓 𝑥 = 𝑥^2 − 8𝑥 + 16

B) 𝑓 𝑥 = 𝑥^2 − 𝑥 + 1