/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 For the following quadratic func... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 12

For the following quadratic functions in vertex form, \(f(x)=a(x-h)^{2}+k,\) determine the values for \(a, h,\) and \(k\) Then compare each to \(f(x)=x^{2},\) and identify which constants represent a stretch/compression factor, or a shift in a particular direction. a. \(p(x)=5(x-4)^{2}-2\) b. \(g(x)=\frac{1}{3}(x+5)^{2}+4\) c. \(h(x)=-0.25\left(x-\frac{1}{2}\right)^{2}+6\) d. \(k(x)=-3(x+4)^{2}-3\)

Short Answer

Expert verified
Values: *p(x)*: \(a=5, h=4, k=-2\) *g(x)*: \(a=\frac{1}{3}, h=-5, k=4\) *h(x)*: \(a=-0.25, h=\frac{1}{2}, k=6\) *k(x)*: \(a=-3, h=-4, k=-3\) Transforms are stretch/compression by \(a\), horizontal shift by \(h\), vertical shift by \(k\).

Step by step solution

01

Determine the values of a, h, and k

Identify the coefficients for each function in the vertex form formula, which is given by: \[ p(x) = 5(x-4)^2 -2 \] \[ g(x) = \frac{1}{3}(x+5)^2 +4 \] \[ h(x) = -0.25\left(x-\frac{1}{2}\right)^2 +6 \] \[ k(x) = -3(x+4)^2 -3 \] We extract the values for each function: *p(x)*: \( a = 5 \), \( h = 4 \), \( k = -2 \) *g(x)*: \( a = \frac{1}{3} \), \( h = -5 \), \( k = 4 \) *h(x)*: \( a = -0.25 \), \( h = \frac{1}{2} \), \( k = 6 \) *k(x)*: \( a = -3 \), \( h = -4 \), \( k = -3 \)
02

Compare each function to f(x)=x^2

The base function is given by: \[ f(x) = x^2 \] The comparison involves identifying the nature of each transformation For *p(x)*: \( p(x) = 5(x-4)^2 -2 \) For *g(x)*: \( g(x) = \frac{1}{3}(x+5)^2 +4 \) For *h(x)*: \( h(x) = -0.25\left(x-\frac{1}{2}\right)^2 + 6 \) For *k(x)*: \( k(x) = -3(x+4)^2 -3 \)
03

Identify stretch/compression and shifts

Observe the coefficients of each function to identify the transformations: **Stretch/Compression:** This is related to the coefficient \(a\). * For *p(x)*: \( a = 5 \) indicates a vertical stretch by a factor of 5. * For *g(x)*: \( a = \frac{1}{3} \) indicates a vertical compression by a factor of \( \frac{1}{3} \). * For *h(x)*: \( a = -0.25 \) indicates a vertical compression by a factor of 0.25, along with a reflection across the x-axis. * For *k(x)*: \( a = -3 \) indicates a vertical stretch by a factor of 3, along with a reflection across the x-axis.**Horizontal Shift:** This is related to \(h\) value. * For *p(x)*: \( h = 4 \) indicates a shift 4 units to the right. * For *g(x)*: \( h = -5 \) indicates a shift 5 units to the left. * For *h(x)*: \( h = \frac{1}{2} \) indicates a shift \( \frac{1}{2} \) unit to the right. * For *k(x)*: \( h = -4 \) indicates a shift 4 units to the left. **Vertical Shift:** The value \(k\) determines the vertical shift direction. * For *p(x)*: \( k = -2 \) indicates a shift 2 units downward. * For *g(x)*: \( k = 4 \) indicates a shift 4 units upward. * For *h(x)*: \( k = 6 \) indicates a shift 6 units upward. * For *k(x)*: \( k = -3 \) indicates a shift 3 units downward.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex Form
The vertex form of a quadratic function is given by:
\( f(x) = a(x-h)^2 + k \)
This form is very useful because it directly shows the vertex of the parabola, which is at the point \( (h, k) \). The coefficient \( a \) determines whether the parabola opens upwards or downwards as well as how 'stretched' or 'compressed' it is.
Quadratic Transformations
Quadratic functions can be transformed in several ways:
  • Stretching or compressing
  • Shifting horizontally or vertically
  • Reflecting across the x-axis
The vertex form helps to quickly identify these transformations through the values of \( a \), \( h \), and \( k \).
Vertical Stretch/Compression
The value of \( a \) in the vertex form \( f(x) = a(x-h)^2+k \) controls the vertical stretch or compression of the function:
  • If \( |a| > 1 \), the function is stretched vertically.
  • If \( |a| < 1 \), the function is compressed vertically.
  • If \( a < 0 \), the function is reflected across the x-axis.
Examples:
  • For \( p(x) = 5(x-4)^2 - 2 \), \( a = 5 \) which means the graph is vertically stretched by a factor of 5.
  • For \( g(x) = \frac{1}{3}(x+5)^2 + 4 \), \( a = \frac{1}{3} \) which means the graph is vertically compressed by a factor of \( \frac{1}{3} \).
Horizontal Shift
The value of \( h \) in the vertex form \( f(x) = a(x-h)^2 + k \) controls the horizontal shift of the function:
  • If \( h > 0 \), the function shifts \( h \) units to the right.
  • If \( h < 0 \), the function shifts \( |h| \) units to the left.
Examples:
  • For \( p(x) = 5(x-4)^2 -2 \), \( h = 4 \) so the graph is shifted 4 units to the right.
  • For \( g(x) = \frac{1}{3}(x+5)^2 + 4 \), \( h = -5 \) so the graph is shifted 5 units to the left.
Vertical Shift
The value of \( k \) in the vertex form \( f(x) = a(x-h)^2+k \) controls the vertical shift of the function:
  • If \( k > 0 \), the function shifts \( k \) units upwards.
  • If \( k < 0 \), the function shifts \( |k| \) units downwards.
Examples:
  • For \( p(x)= 5(x-4)^2-2 \), \( k = -2 \) so the graph is shifted 2 units downward.
  • For \( g(x) = \frac{1}{3}(x+5)^2 + 4 \), \( k = 4 \) so the graph is shifted 4 units upward.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A shot-put athlete releases the shot at a speed of 14 meters per second, at an angle of 45 degrees to the horizontal (ground level). The height \(y\) (in meters above the ground) of the shot is given by the function $$ y=2+x-\frac{1}{20} x^{2} $$ where \(x\) is the horizontal distance the shot has traveled (in meters). a. What was the height of the shot at the moment of release? b. How high is the shot after it has traveled 4 meters horizontally from the release point? 16 meters? c. Find the highest point reached by the shot in its flight. d. Draw a sketch of the height of the shot and indicate how far the shot is from the athlete when it lands.

a. If the degree of a polynomial is odd, then at least one of its zeros must be real. Explain why this is true. b. Sketch a polynomial function that has no real zeros and whose degree is: i. 2 ii. 4 c. Sketch a polynomial function of degree 3 that has exactly: i. One real zero ii. Three real zeros d. Sketch a polynomial function of degree 4 that has exactly two real zeros.

If we know the radius and depth of a parabolic reflector, we also know where the focus is. a. Find a generic formula for the focal length \(f\) of a parabolic reflector expressed in terms of its radius \(R\) and depth \(D\). The focal length \(\left|\frac{1}{4 a}\right|\) is the distance between the vertex and the focal point. Assume \(a>0\). b. Under what conditions does \(f=D ?\)

(Graphing program required.) At low speeds an automobile engine is not at its peak efficiency; efficiency initially rises with speed and then declines at higher speeds. When efficiency is at its maximum, the consumption rate of gas (measured in gallons per hour) is at a minimum. The gas consumption rate of a particular car can be modeled by the following equation, where \(G\) is the gas consumption rate in gallons per hour and \(M\) is speed in miles per hour: \(G=0.0002 M^{2}-0.013 M+1.07\) a. Construct a graph of gas consumption rate versus speed. Estimate the minimum gas consumption rate from your graph and the speed at which it occurs. b. Using the equation for \(G,\) calculate the speed at which the gas consumption rate is at its minimum. What is the minimum gas consumption rate? c. If you travel for 2 hours at peak efficiency, how much gas will you use and how far will you go? d. If you travel at \(60 \mathrm{mph}\), what is your gas consumption rate? How long does it take to go the same distance that you calculated in part (c)? (Recall that travel distance = speed \(\times\) time traveled.) How much gas is required for the trip? e. Compare the answers for parts (c) and (d), which tell you how much gas is used for the same-length trip at two different speeds. Is gas actually saved for the trip by traveling at the speed that gives the minimum gas consumption rate? f. Using the function \(G,\) generate data for gas consumption rate measured in gallons per mile by completing the following table. Plot gallons per mile (on the vertical axis) vs. miles per hour (on the horizontal axis). At what speed is gallons per mile at a minimum? g. Add a fourth column to the data table. This time compute miles/gal \(=\mathrm{mph} /(\mathrm{gal} / \mathrm{hr}) .\) Plot miles per gallon vs. miles per hour. At what speed is miles per gallon at a maximum? This is the inverse of the preceding question; we are normally used to maximizing miles per gallon instead of minimizing gallons per mile. Does your answer make sense in terms of what you found for parts (b) and (f)?

For each of the following quadratic functions, find the vertex \((h, k)\) and determine if it represents the maximum or minimum of the function. a. \(f(x)=-2(x-3)^{2}+5\) c. \(f(x)=-5(x+4)^{2}-7\) b. \(f(x)=1.6(x+1)^{2}+8\) d. \(f(x)=8(x-2)^{2}-6\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.