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Chapter 8: Problem 5

Identify the stretch/compression factor and the vertex for each of the following. a. \(y_{1}=0.3(x-1)^{2}+8\) c. \(y_{3}=0.01(x+20)^{2}\) b. \(y_{2}=30 x^{2}-11\) d. \(y_{4}=-6 x^{2}+12 x\)

Short Answer

Expert verified
y1: compression by 0.3, vertex (1, 8). y2: stretch by 30, vertex (0, -11). y3: compression by 0.01, vertex (-20, 0). y4: stretch by 6, vertex (1, 6), reflected.

Step by step solution

01

Title - Identify stretch/compression factor and vertex for y1

Given equation: \(y_{1}=0.3(x-1)^{2}+8\)1. The coefficient of \( (x-1)^2 \) is 0.3, which means there is a vertical compression by a factor of 0.3.2. The vertex form of a parabola is given by \( y=a(x-h)^2+k \), where \( (h, k) \) is the vertex. Here, \( h = 1 \) and \( k = 8 \).3. Therefore, the vertex is \( (1, 8) \).
02

Title - Identify stretch/compression factor and vertex for y2

Given equation: \(y_{2}=30 x^{2}-11\)1. The coefficient of \( x^2 \) is 30, which means there is a vertical stretch by a factor of 30.2. The vertex form does not show any horizontal shift, so \( h = 0 \) and \( k = -11 \).3. Therefore, the vertex is \( (0, -11) \).
03

Title - Identify stretch/compression factor and vertex for y3

Given equation: \(y_{3}=0.01(x+20)^{2}\)1. The coefficient of \( (x+20)^2 \) is 0.01, which means there is a vertical compression by a factor of 0.01.2. The vertex form shows that \( h = -20 \) and \( k = 0 \).3. Therefore, the vertex is \( (-20, 0) \).
04

Title - Identify stretch/compression factor and vertex for y4

Given equation: \(y_{4}=-6 x^{2}+12 x\)1. Rewrite the equation in vertex form by completing the square.\[ y = -6(x^{2} - 2x) \]\[ y = -6((x-1)^{2} - 1) \]\[ y = -6(x-1)^{2} + 6 \]2. The coefficient of \( (x-1)^2 \) is -6, which means there is a vertical stretch by a factor of 6 and a reflection over the x-axis.3. The vertex form shows that \( h = 1 \) and \( k = 6 \).4. Therefore, the vertex is \( (1, 6) \).

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Key Concepts

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

Vertex Form of a Parabola
The vertex form of a parabola provides an easy way to identify the vertex of the parabola directly from its equation. The vertex form is written as:\( y = a(x - h)^2 + k \)Here, \( (h, k) \) is the vertex of the parabola. By adjusting the values of \( h \) and \( k \), you can shift the parabola horizontally and vertically on the coordinate plane.

For example, the equation \( y = 0.3(x - 1)^2 + 8 \) has its vertex at \( (1, 8) \). You can see this because the term inside the parenthesis affects the x-coordinate (\( h \)) and the term added or subtracted outside the parenthesis affects the y-coordinate (\( k \)).

Recognizing vertex form is key to quickly determining the vertex and making transformations easier to understand.Key Points:
  • The coefficient \( a \) affects the vertical stretch or compression.
  • The value \( h \) moves the parabola left or right.
  • The value \( k \) moves the parabola up or down.
Vertical Stretch/Compression
The coefficient \( a \) in the vertex form equation \( y = a(x - h)^2 + k \) is crucial because it affects the width and direction of the parabola.

When \( |a| > 1 \), the parabola undergoes a vertical stretch, meaning it becomes narrower. When \( 0 < |a| < 1 \), the parabola undergoes a vertical compression, making it wider. If \( a \) is negative, the parabola opens downward instead of upward.

For instance, in the equation \( y = 30x^2 - 11 \), the coefficient \( 30 \) stretches the parabola by a factor of 30, making it very narrow. Conversely, in \( y = 0.01(x + 20)^2 \), the coefficient \( 0.01 \) compresses the parabola, making it much wider.Key Points:
  • Positive \( a \): parabola opens upward.
  • Negative \( a \): parabola opens downward.
  • \( |a| > 1 \): narrow parabola (vertical stretch).
  • \( 0 < |a| < 1 \): wide parabola (vertical compression).
Vertex Calculation
Calculating the vertex for a parabola in standard form \( y = ax^2 + bx + c \) can be done by converting it to vertex form. This often involves completing the square. Consider the example given in the exercise:For the equation \( y = -6x^2 + 12x \), we can rewrite it step by step:1. Factor out the coefficient of \( x^2 \):\[ y = -6(x^2 - 2x) \]2. Complete the square inside the parentheses:\[ y = -6((x - 1)^2 - 1) \]3. Distribute the \( -6 \) and adjust the constant term:\[ y = -6(x - 1)^2 + 6 \]From this, the vertex is \( (1, 6) \).Key Steps:
  • Factor out the coefficient of \( x^2 \).
  • Complete the square.
  • Adjust the constant term.

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Most popular questions from this chapter

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.

a. In economics, revenue \(R\) is defined as the amount of money derived from the sale of a product and is equal to the number \(x\) of units sold times the selling price \(p\) of each unit. What is the equation for revenue? b. If the selling price is given by the equation \(p=-\frac{1}{10} x+20,\) express revenue \(R\) as a function of the number \(x\) of units sold. c. Using technology, plot the function and estimate the number of units that need to be sold to achieve maximum revenue. Then estimate the maximum revenue.

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. \(f(x)=8(x-2)^{2}-6\)

(Graphing program optional.) Create a quadratic function in the vertex form \(y=a(x-h)^{2}+k,\) given the specified values for \(a\) and the vertex \((h, k) .\) Then rewrite the function in the standard form \(y=a x^{2}+b x+c .\) If available, use technology to check that the graphs of the two forms are the same. a. \(a=1,(h, k)=(2,-4)\) c. \(a=-2,(h, k)=(-3,1)\) b. \(a=-1,(h, k)=(4,3)\) d. \(a=\frac{1}{2},(h, k)=(-4,6)\)

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.
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