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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

Let \((h, k)\) be the coordinates of the vertex of a parabola. Then \(h\) is equal to the average of the two real zeros of the function (if they exist). For parts (a) and (b) use this to find \(h\), and then construct an equation in vertex form, \(y=a(x-h)^{2}+k\). a. A parabola with \(x\) -intercepts of 4 and 8 , and a \(y\) -intercept of 32 b. A parabola with \(x\) -intercepts of -3 and \(1,\) and a \(y\) -intercept of -1 c. Can you find the equation of a parabola knowing only its \(x\) -intercepts? Explain.

Let \(f(x)=x^{3}\). a. Write the equation for the new function \(g(x)\) that results from each of the following transformations of \(f(x)\). Explain in words the effect of the transformations. i. \(f(-x)\) iii. \(f(x+2)\) ii. \(-2 f(x)-1\) iv. \(-f(-x)\) b. Sketch by hand the graph of \(f(x)\) and each function in part (a).

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

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 \(f=D ?\)
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