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You have learned how to move a parabola around a set of axes, write equations, sketch graphs, and model

situations using transformed parabolas.  The graph of y = x2 is called the parent graph for the family ofparabolas because every other parabola is a transformation of that one graph.

2-25.  In this investigation you will use what you have learned abouttransforming the graph of y = x2 to transform five other parent graphs. In fact, your team will figure out how to use what you have learned

to transform the graph ofany function!

 

Your Task: As a team, determine how you can translate the graph ofany function left, right, up, and down; how you can stretch or

compress it vertically; and how you can reflect it over the x‑axis.  Each team member should investigateone of the following parent functions: y =x3, y =  , y =  , y =   , and y = bx.  (If you are

investigating y = bx, your teacher will give you a value to use for b.)

Remember that to investigate completely, you should include a complete description of your graph.

Graph and write an equation to demonstrate each transformation you find.

Finally, write an equation in graphing form for your family of graphs.

How can we transform a parabola?

How can we use our ideas about transforming parabolas to transform other functions?

What changes can we make to the equation?

2-26.  First, investigate your parent graph.

a. Graph your equation on a full sheet of graph paper.

b. As a team, place your parent graphs into the middle of your workspace.  For each graph, identify

the domain and range and label any important points or asymptotes.

2-27.  For your parent graph:

a. Write and graph an equation that will translate your parent graph left or right.

b. Write and graph an equation that will translate your parent graph up or down.

c. Write and graph an equation that will stretch or compress your parent graph vertically.

d. Write and graph an equation that will reflect your parent graph across the x‑axis.

2-28.  One way of writing an equation for a parabola is to use graphing form: y = a(x – h)2 + k.  Thisequation tells you how to transform the parent graph, y = x2, to get any other parabola.

a. Explain what each parameter (a, h, and k) represents for the graph of a parabola.  Explore this usingthe 2-28 Student eTool (Desmos).

b. As a team, write equations in graphing form for each given parent function.  Be ready to explain

how your equations in graphing form work; that is, tell what effect each parameter has on the

vertical stretch, horizontal shift, vertical shift, and orientation of the graph.

2-29.  As a team, organize your work into a large poster that shows clearly:

Each parent graph you worked with,

Examples of each transformation you found, and

Each equation in graphing form.

Use colors, arrows, and shading to show all of the connections you discover.  Then add the following

problems for other teams to solve:

Show the graph of a function in your family for which other teams need to write the equation.

Give an equation of a function in your family that other teams will graph.

Rewriting Radical ExpressionsBefore calculators were readily available, people found it convenient to rewrite square root

expressions in a simplified form to make it easier to perform calculations, combine, and compare

them.  A square root is simplified when there are no more perfect square factors (square numbers

such as 4, 25, and 81) under the radical sign.

Example:   can be rewritten as   ·  , or as  ·   and you still get the same result when you simplify

completely.  Verify with your calculator that both 6

 and   ≈ 8.485.

Example:   + 

To add terms with radicals, the radicals need to be alike. 

Verify your simplification with your calculator.

It is difficult to estimate the value of a number with a

radical in the denominator, and it is also difficult to

combine it with or compare it to other numbers.  For

these reasons, it is sometimes helpful to rationalize the denominator so that no radical remains inthe denominator.

Example: 

First, multiply the numerator and denominator by the radical in the

denominator.  Since   = 1, this does notchange the value of the expression. 

After multiplying, notice that the denominator no longer has a radical, since   ·   = 2.

Often, the product can be further simplified.

 

2-30.  Maura is deciding which hose to use to water her outdoorplants.  Maura notices that the water coming out of her garden hoses

follows a parabolic path.

When Maura uses her green garden hose the greatest height the water

reaches is 8 feet, and it lands on the plants 10 feet from where she is

standing.  Both the nozzle of the hose and the top of the flowers are 4

feet above the ground.

Maura has already determined that when she has the water on full blast, the water from the red hose

follows the pathy = –(x – 3)2 + 7. Homework Help ✎a. Which hose will throw the water higher?

 

b. Write an equation that models the path of the water from Maura’s green hose.

 

c. What domain and range make sense for the water from Maura’s green hose?

2-31.  Draw the graph of y = 2x2 + 3x + 1. 2-31 HW eTool (Desmos). Homework Help ✎

a. What are the x‑ and y‑intercepts? 

b. Where is the line of symmetry of this parabola?  Write its equation. 

 

c. What are the coordinates of the vertex?

2-32.  Change one term in the equation in problem 2‑31 so that the parabola has onlyone x‑intercept.  Homework Help ✎2-33.  Simplify each expression.  Refer to the Math Notes box in this lessonfor help.  Homework Help ✎

a.

 

b.

 

c.

2-34.  Below are two situations that can be described using exponential functions.  They represent a smallsample of the situations where quantities grow or decay by a constant percentage over equal periods of

time.  For each situation:  Homework Help ✎What is an appropriate unit of time (such as days, weeks, years)?

What is the multiplier?

What is the initial value?

 Write an exponential equation in the form f(x) = abxthat represents the situation.

a. The value of a car with an initial purchase price of $12,250 depreciates by 11% per year.

b. An investment of $1000 earns 6% annual interest, compounded monthly.

2-35.  Harvey’s Espresso Express, a drive-through coffee stop, is famous for itsdelicious house coffee, a blend of Colombian and Mocha Java beans.  Their

archrival, Jojo’s Java, sent a spy to steal their ratio for blending beans.  The spy

returned with a torn part of an old receipt that showed only the total number of

pounds and the total cost, 18 pounds for $92.07.  At first Jojo was angry, but then

he realized that he knew the price per pound of each kind of coffee ($4.89 for

Colombian and $5.43 for Mocha Java).  Show how he could use equations to

figure out how many pounds of each type of beans Harvey’s used to make its

house coffee.  Homework Help ✎2-36.  Match each graph below with its domain.  Then state the range of eachgraph.  Homework Help ✎

a.         D: all real numbers b.         D: x > –2 c.         D: x ≤ 3

 

2-37.  Sketch a graph and draw the line of symmetry for the function y = 2(x – 4)2 – 3.  What is theequation of the line of symmetry? Homework Help ✎  

2-38.  The Gross National Product (GNP) of the United States in 1960 was about 2.19 · 1012 dollars. Suppose it increased at a rate of 3.17% per year.  Use this information to answer each of the questions

below.  Homework Help ✎a. What was the predicted GNP in 1989?

b. Write an equation to model the GNP t years after 1960. 

c. What assumptions did you make in creating your model?  How accurate do you think it is?

2-39.  Simplify each expression.  Refer to the Math Notes box in this lesson for help. Homework Help ✎

    a.   b.   c.    +  d.    + 

2-40.   In the diagram at right,ΔABC ∼ ΔAED. Homework Help ✎ 

a. Solve for x.

b. Calculate the perimeter of  ΔAED.

 

 

2-41.  What is the distance from the point (5, 10) to the liney = –x + 5? Hint: Draw a perpendicular line.  Homework Help ✎ 

2-42.  Solve for z in each equation.  Homework Help ✎

a. 2z = 23

b. 4z = 8

c. 3z = 812

d. 5(z+1)/3 = 251/z

2-43.  The vertex of a parabola, point (h, k), locates its position on the coordinate graph.  The vertex thusserves as a locator point for a parabola.  Other families of functions that you will be investigating in thiscourse will also have locator points.  These points have different names, but the same purpose for eachdifferent type of graph.  These points help you place the graph on the axes.

Sketch graphs for both of the following functions.  On each graph, label the locator point.  HomeworkHelp ✎  

a. y = 3x2 + 5

b. f(x) = –(x – 3)2 – 7

2-44.  Write a possible equation for each of these graphs.  Assume that all axes are scaled by one unit. Check your equations on a graphing calculator. Homework Help ✎ 

2-45.  By mistake, Jim graphed y = x3 – 4x instead of y = x3– 4x + 6.  What should he do to his graph toget the correct one?  Homework Help ✎

2-46.  Where do the following pairs of lines intersect?  Homework Help ✎a. y = 5x – 2y = 3x + 18

b. y = x – 42x + 3y = 17 

2-47. The right triangle shown at right has a height (BC) of 12.0 cm, and its area is60.0 square cm.  What are m∠Band the length of the hypotenuse?  Homework Help✎2-48.  Simplify each of the following expressions.  Be sure that your answer has no

negative or fractional exponents.  Homework Help ✎

a.

b. x−2y−4

c. (2x)−2(16x2y)1/2

2-49.  Solve each equation.  Be sure to check each solution using the original equation.  Homework Help✎

a. (n + 4) + n(n + 2) + n = 0 

b.  = x + 3

2-50.  For each equation, what input value will produce thesmallest possible output?  (In other words,what is thex‑value of the lowest point on the graph?)  What is the input value that givesthe largest possible output (or thex‑value of the highest point on the graph)?  Homework Help ✎

a. y = (x – 2)2 

b. y = x2 + 2 

c. y = (x + 3)2 

d. y = –x2 + 5