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Parabolas & Quadratic Equations: Vertices, Graphing, and Solving Max/Min Problems, Study notes of 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 โ‰  1?
  • How do you graph a quadratic function?
  • What is the formula for the vertex of a quadratic function?

Typology: Study notes

2021/2022

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9.6 More About
Parabolas & Their Applications
OBJECTIVES:
โ€ขFind the vertex of the vertical parabola.
โ€ขGraph a quadratic function.
โ€ขUse the discriminant to find the number of x-
intercepts of a parabola with a vertical axis.
โ€ขUse quadratic functions to solve problems
involving maximum or minimum values.
โ€ขGraph parabolas with horizontal axes.
Written by: Cindy Alder
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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

๏ƒ˜ We can tell a lot about a quadratic equation when it is in the form:

๐’‡ ๐’™ = ๐’‚ ๐’™ โˆ’ ๐’‰ ๐Ÿ^ + ๐’Œ

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

axis of symmetry x = h

Vertex ( h, k )

Complete the Square when ๐’‚ = ๐Ÿ

๐’‡ ๐’™ = ๐’™๐Ÿ^ โˆ’ ๐Ÿ’๐’™ + ๐Ÿ“

  • Group the variable terms. Slide the constant over to โ€œdeal with laterโ€.
  • Find the โ€œmagicโ€ number needed to complete the square.
  • Add and subtract the โ€œmagicโ€ number to the equation.
  • Factor and combine like terms.

Find the Vertex of the Graph of

๐’‡ ๐’™ = ๐’™๐Ÿ^ + ๐Ÿ”๐’™ + ๐Ÿ๐Ÿ’

Complete the Square when ๐’‚ โ‰  ๐Ÿ

๐’‡ ๐’™ = โˆ’๐Ÿ‘๐’™๐Ÿ^ + ๐Ÿ”๐’™ โˆ’ ๐Ÿ
  • Group the variable terms. Slide the constant over to โ€œdeal with laterโ€.
  • Factor out the coefficient on ๐’™๐Ÿ^ from the variable terms.
  • Find the โ€œmagicโ€ number needed to complete the square.
  • Add the โ€œmagicโ€ number to the equation and
  • Subtract the product of the โ€œmagicโ€ number and the coefficient you factored out from the equation.
  • Factor and combine like terms.

Find the Vertex of the Graph of

๐’‡ ๐’™ = ๐Ÿ’๐’™๐Ÿ^ โˆ’ ๐Ÿ‘๐Ÿ๐’™ + ๐Ÿ“๐Ÿ–

The Vertex Formula

๐’‡ ๐’™ = ๐’‚๐’™๐Ÿ^ + ๐’ƒ๐’™ + ๐’„

Vertex Formula

โˆ’๐’ƒ ๐Ÿ๐’‚

โˆ’๐’ƒ ๐Ÿ๐’‚ ๐’š = ๐’‡^ ๐’•๐’‰๐’† ๐’—๐’‚๐’๐’–๐’† ๐’š๐’๐’– ๐’ˆ๐’๐’• ๐’‡๐’๐’“ ๐’™

Find the vertex using the vertex formula.

๏ƒ˜ ๐’‡ ๐’™ = ๐’™๐Ÿ^ + ๐Ÿ–๐’™ + ๐Ÿ๐ŸŽ

Graphing a Quadratic Function

๏ƒ˜ STEPS ๏ƒ˜ Does the graph open up or down? ๏ƒ˜ Find the vertex. ๏ƒ˜ Vertex formula ๏ƒ˜ Complete the square ๏ƒ˜ Find any intercepts. ๏ƒ˜ y- intercept: set ๐‘ฅ = 0 ๏ƒ˜ x- intercept: set ๐‘ฆ = 0 ๏ƒ˜ Complete the graph. ๏ƒ˜ Plot points found. ๏ƒ˜ 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:

๐’‡ ๐’™ = ๐’™๐Ÿ^ โˆ’ ๐Ÿ”๐’™ + ๐Ÿ“

๏‚จ Step 1 - Does the graph open up or down?

  • The graph will open up because a is positive. ๏‚จ Step 2 - Find the vertex.

+ +

๏‚จ Step 4 - Complete the graph. ๏ƒ˜ Plot points found. ๏ƒ˜ Vertex: 3, โˆ’ ๏ƒ˜ y- intercept: 0, ๏ƒ˜ x- intercepts: 5,0 ๐‘Ž๐‘›๐‘‘ (1,0)

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

Graph the Quadratic Equation (continued)

๐’‡ ๐’™ = ๐’™๐Ÿ^ โˆ’ ๐Ÿ”๐’™ + ๐Ÿ“

Graph the Function ๐’‡ ๐’™ = ๐’™๐Ÿ^ โˆ’ ๐’™ โˆ’ ๐Ÿ”

Using the Discriminant

  • Find the discriminant and use it to determine the number of

x -intercepts of the graph of each quadratic function.

A) ๐‘“ ๐‘ฅ = โˆ’3๐‘ฅ^2 โˆ’ ๐‘ฅ + 2
C) ๐‘“ ๐‘ฅ = ๐‘ฅ^2 โˆ’ 8๐‘ฅ + 16
B) ๐‘“ ๐‘ฅ = ๐‘ฅ^2 โˆ’ ๐‘ฅ + 1

Parabola with Horizontal Axis

  • The graph of

๐’™ = ๐’‚๐’š๐Ÿ^ + ๐’ƒ๐’š + ๐’„ or ๐’™ = ๐’‚ ๐’š โˆ’ ๐’Œ ๐Ÿ^ + ๐’‰

is a parabola with vertex (โ„Ž, ๐‘˜) and the horizontal line ๐‘ฆ = ๐‘˜ as axis of symmetry. The graph opens to the right if ๐‘Ž > 0 and to the left if ๐‘Ž < 0.