Given a function , which of the following represents a

In Exercises 1-6, write the expression as a binomial squared. \(x^{2}+2 x+1\)

In Exercises 1-6, write the expression as a binomial squared. \(x^{2}-6 x+9\)

In Exercises 1-6, write the expression as a binomial squared. \(x^{2}+12 x+36\)

In Exercises 1-6, write the expression as a binomial squared. \(4 x^{2}+4 x+1\)

In Exercises 1-6, write the expression as a binomial squared. \(x^{2}-5 x+\frac{25}{4}\)

In Exercises 1-6, write the expression as a binomial squared. \(4 x^{2}-20 x+25\)

In Exercises 7-10, perform the indicated operations and simplify. \((x-2)^{2}+3(x-2)+4\)

In Exercises 7-10, perform the indicated operations and simplify. \(2(x+3)^{2}-5(x+3)-2\)

In Exercises 7-10, perform the indicated operations and simplify. \((x-1)^{3}+3(x-1)^{2}-3(x-1)\)

In Exercises 7-10, perform the indicated operations and simplify. \(2(x+1)^{3}-6(x+1)^{2}+6(x+1)-2\)

In Exercises 1-8, describe how the graph of y = x2 can be transformed to the graph of the given equation. \(y=x^{2}-3\)

In Exercises 1-8, describe how the graph of y = x2 can be transformed to the graph of the given equation. \(y=x^{2}+5.2\)

In Exercises 1-8, describe how the graph of y = x2 can be transformed to the graph of the given equation. \(y=(x+4)^{2}\)

In Exercises 1-8, describe how the graph of y = x2 can be transformed to the graph of the given equation. \(y=(x-3)^{2}\)

In Exercises 1-8, describe how the graph of y = x2 can be transformed to the graph of the given equation. \(y=(100-x)^{2}\)

In Exercises 1-8, describe how the graph of y = x2 can be transformed to the graph of the given equation. \(y=x^{2}-100\)

In Exercises 1-8, describe how the graph of y = x2 can be transformed to the graph of the given equation. \(y=(x-1)^{2}+3\)

In Exercises 1-8, describe how the graph of y = x2 can be transformed to the graph of the given equation. \(y=(x+50)^{2}-279\)

In Exercises 9-12, describe how the graph of \(y=\sqrt{x}\) can be transformed to the graph of the given equation. \(y=-\sqrt{x}\)

In Exercises 9-12, describe how the graph of \(y=\sqrt{x}\) can be transformed to the graph of the given equation. \(y=\sqrt{x-5}\)

In Exercises 9-12, describe how the graph of \(y=\sqrt{x}\) can be transformed to the graph of the given equation. \(y=\sqrt{-x}\)

In Exercises 9-12, describe how the graph of \(y=\sqrt{x}\) can be transformed to the graph of the given equation. \(y=\sqrt{3-x}\)

In Exercises 13-16, describe how the graph of \(y=x^{3}\) can be transformed to the graph of the given equation. \(y=2 x^{3}\)

In Exercises 13-16, describe how the graph of \(y=x^{3}\) can be transformed to the graph of the given equation. \(y=(2 x)^{3}\)

In Exercises 13-16, describe how the graph of \(y=x^{3}\) can be transformed to the graph of the given equation. \(y=(0.2 x)^{3}\)

In Exercises 13-16, describe how the graph of \(y=x^{3}\) can be transformed to the graph of the given equation. \(y=0.3 x^{3}\)

In Exercises 17-20, describe how to transform the graph of f into the graph of g \(f(x)=\sqrt{x+2} \text { and } g(x)=\sqrt{x-4}\)

In Exercises 17-20, describe how to transform the graph of f into the graph of g \(f(x)=(x-1)^{2} \text { and } g(x)=-(x+3)^{2}\)

In Exercises 17-20, describe how to transform the graph of f into the graph of g \(f(x)=(x-2)^{3} \text { and } g(x)=-(x+2)^{3}\)

In Exercises 17-20, describe how to transform the graph of f into the graph of g \(f(x)=|2 x| \text { and } g(x)=4|x|\)

In Exercises 21-24, sketch the graphs of f, g, and h by hand. Support your answers with a grapher. \(\begin{array}{l}f(x)=(x+2)^{2} \\g(x)=3 x^{2}-2 \\h(x)=-2(x-3)^{2}\end{array}\)

In Exercises 21-24, sketch the graphs of f, g, and h by hand. Support your answers with a grapher. \(\begin{array}{l}f(x)=x^{3}-2 \\g(x)=(x+4)^{3}-1 \\h(x)=2(x-1)^{3}\end{array}\)

In Exercises 21-24, sketch the graphs of f, g, and h by hand. Support your answers with a grapher. \(\begin{array}{l}f(x)=\sqrt[3]{x+1} \\g(x)=2 \sqrt[3]{x}-2 \\h(x)=-\sqrt[3]{x-3}\end{array}\)

In Exercises 21-24, sketch the graphs of f, g, and h by hand. Support your answers with a grapher. \(\begin{array}{l}f(x)=-2|x|-3 \\g(x)=3|x+5|+4 \\h(x)=|3 x|\end{array}\)

In Exercises 25-28, the graph is that of a function y = f(x) that can be obtained by transforming the graph of \(y=\sqrt{x}\). Write a formula for the function f.

In Exercises 25-28, the graph is that of a function y = f(x) that can be obtained by transforming the graph of \(y=\sqrt{x}\). Write a formula for the function f.

In Exercises 25-28, the graph is that of a function y = f(x) that can be obtained by transforming the graph of \(y=\sqrt{x}\). Write a formula for the function f.

In Exercises 25-28, the graph is that of a function y = f(x) that can be obtained by transforming the graph of \(y=\sqrt{x}\). Write a formula for the function f.

In Exercises 29-32, find the equation of the reflection of across (a) the x-axis and (b) the y-axis. \(f(x)=x^{3}-5 x^{2}-3 x+2\)

In Exercises 29-32, find the equation of the reflection of across (a) the x-axis and (b) the y-axis. \(f(x)=2 \sqrt{x+3}-4\)

In Exercises 29-32, find the equation of the reflection of across (a) the x-axis and (b) the y-axis. \(f(x)=\sqrt[3]{8 x}\)

In Exercises 29-32, find the equation of the reflection of across (a) the x-axis and (b) the y-axis. \(f(x)=3|x+5|\)

Reflecting Odd Functions Prove that the graph of an odd function is the same when reflected across the x-axis as it is when reflected across the y-axis.

Reflecting Odd Functions Prove that if an odd function is reflected about the y-axis and then reflected again about the x-axis, the result is the original function.

Exercises 35-38 refer to the graph of y =f(x) shown at the top of the next column. In each case, sketch a graph of the new function. \(y=|f(x)|\)

Exercises 35-38 refer to the graph of y =f(x) shown at the top of the next column. In each case, sketch a graph of the new function. \(y=f(|x|)\)

Exercises 35-38 refer to the graph of y =f(x) shown at the top of the next column. In each case, sketch a graph of the new function. \(y=-f(|x|)\)

Exercises 35-38 refer to the graph of y =f(x) shown at the top of the next column. In each case, sketch a graph of the new function. \(y=|f(|x|)|\)

In Exercises 39-42, transform the given function by (a) a vertical stretch by a factor of 2, and (b) a horizontal shrink by a factor of 1/3 \(f(x)=x^{3}-4 x\)

In Exercises 39-42, transform the given function by (a) a vertical stretch by a factor of 2, and (b) a horizontal shrink by a factor of 1/3 \(f(x)=|x+2|\)

In Exercises 39-42, transform the given function by (a) a vertical stretch by a factor of 2, and (b) a horizontal shrink by a factor of 1/3 \(f(x)=x^{2}+x-2\)

In Exercises 39-42, transform the given function by (a) a vertical stretch by a factor of 2, and (b) a horizontal shrink by a factor of 1/3 \(f(x)=\frac{1}{x+2}\)

In Exercises 43-46, describe a basic graph and a sequence of transformations that can be used to produce a graph of the given function. \(y=2(x-3)^{2}-4\)

In Exercises 43-46, describe a basic graph and a sequence of transformations that can be used to produce a graph of the given function. \(y=-3 \sqrt{x+1}\)

In Exercises 43-46, describe a basic graph and a sequence of transformations that can be used to produce a graph of the given function. \(y=(3 x)^{2}-4\)

In Exercises 43-46, describe a basic graph and a sequence of transformations that can be used to produce a graph of the given function. y=-2|x+4|+1

In Exercises 47-50, a graph G is obtained from a graph of y by the sequence of transformations indicated. Write an equation whose graph is G. y = x2:a vertical stretch by a factor of 3, then a shift right 4 units.

In Exercises 47-50, a graph G is obtained from a graph of y by the sequence of transformations indicated. Write an equation whose graph is G. y = x2 :a shift right 4 units, then a vertical stretch by a factor of 3.

In Exercises 47-50, a graph G is obtained from a graph of y by the sequence of transformations indicated. Write an equation whose graph is G. y = |x| :a shift left 2 units, then a vertical stretch by a factor of 2, and finally a shift down 4 units.

In Exercises 47-50, a graph G is obtained from a graph of y by the sequence of transformations indicated. Write an equation whose graph is G. y = |x|:a shift left 2 units, then a horizontal shrink by a factor of 1/2, and finally a shift down 4 units.

Exercises 51-54 refer to the function whose graph is shown below. Sketch the graph of y=2+3 f(x+1)

Exercises 51-54 refer to the function whose graph is shown below. Sketch the graph of y=-f(x+1)+1

Exercises 51-54 refer to the function f whose graph is shown below. Sketch the graph of y=f(2 x)

Exercises 51-54 refer to the function f whose graph is shown below. Sketch the graph of \(y=2 f(x-1)+2\)

Writing to Learn Graph some examples to convince yourself that a reflection and a translation can have a different effect when combined in one order than when combined in the opposite order. Then explain in your own words why this can happen.

Writing to Learn Graph some examples to convince yourself that vertical stretches and shrinks do not affect a graphs x-intercepts. Then explain in your own words why this is so.

Celsius vs. Fahrenheit The graph shows the temperature in degrees Celsius in Windsor, Ontario, for one 24-hour period. Describe the transformations that convert this graph to one showing degrees Fahrenheit. [Hint: F(t) = (9/5)C(t) + 32.]

Fahrenheit vs. Celsius The graph shows the temperature in degrees Fahrenheit in Mt. Clemens, Michigan, for one 24-hour period. Describe the transformations that convert this graph to one showing degrees Celsius. [Hint: F(t) = (9/5)C(t) + 32.]

The function y=f(x+3) represents a translation to the right by 3 units of the graph of y=f(x) Justify your answer.

The function y=f(x)-4 represents a translation down 4 units of the graph of y=f(x). Justify your answer.

Given a function f, which of the following represents a vertical stretch by a factor of 3? (A) \(y=f(3 x)\) (B) \(y=f(x / 3)\) (C) \(y=3 f(x)\) (D) \(y=f(x) / 3\) (E) \(y=f(x)+3\)

Given a function , which of the following represents a horizontal translation of 4 units to the right? (A) y=f(x)+4 (B) y=f(x)-4 (C) y=f(x+4) (D) y=f(x-4) (E) y=4 f(x)

Given a function f, which of the following represents a vertical translation of 2 units upward, followed by a reflection across the y-axis? (A) y=f(-x)+2 (B) y=2-f(x) (C) y=f(2-x) (D) y=-f(x-2) (E) y=f(x)-2

Given a function f, which of the following represents reflection across the x-axis, followed by a horizontal shrink by a factor of 1/2? (A) y=-2 f(x) (B) y=-f(x) / 2 (C) y=f(-2 x) (D) y=-f(x / 2) (E) y=-f(2 x)

Table 1.11 shows the (adjusted closing) price of a share of stock in Dell Computer for each month of 2008. (a) Graph price (y) as a function of month (x) as a line graph, connecting the points to make a continuous graph. (b) Explain what transformation you would apply to this graph to produce a graph showing the price of the stock in Japanese yen.

Get with a friend and graph the function y = x2 on both your graphers. Apply a horizontal or vertical stretch or shrink to the function on one of the graphers. Then change the window of that grapher to make the two graphs look the same. Can you formulate a general rule for how to find the window?

The Absolute Value Transformation Graph the function \(f(x)=x^{4}-5 x^{3}+4 x^{2}+3 x+2\) in the viewing window [-5, 5] by [-10, 10]. (Put the equation in .) (a) Study the graph and try to predict what the graph of y = |ƒ(x)| will look like. Then turn Y1 off and graph Y2 = abs (Y1). Did you predict correctly? (b) Study the original graph again and try to predict what the graph of y = ƒ(|x|)will look like. Then turn Y1 off and graph Y2 = Y1(abs (X)). Did you predict correctly? (c) Given the graph of shown below, sketch a graph of y = |g(x)|. (d) Given the graph of y = g(x) shown below, sketch a graph of y = g(|x|).

Parametric Circles and Ellipses Set your grapher to parametric and radian mode and your window as follows: Tmin = 0, Tmax = 7, Tstep = 0.1 Xmin = -4.7, Xmax = 4.7, Xscl = 1 Ymin = -3.1, Ymax = 3.1, Yscl = 1 (a) Graph the parametric equations x = cos t and y = sin t. You should get a circle of radius 1. (b) Use a transformation of the parametric function of x to produce the graph of an ellipse that is 4 units wide and 2 units tall. (c) Use a transformation of both parametric functions to produce a circle of radius 3. (d) Use a transformation of both functions to produce an ellipse that is 8 units wide and 4 units tall. (You will learn more about ellipses in Chapter 8.)