has that by finding the change in y's and the change in the x's. +1 +1 +1 +1. -3 -3 -3 -3. The change in y. change in x.
Name: ________________________________________________Date: _______________________Hour: ___________
Unit 6: Comparing Functions Review Key 1. Determine whether the functions below are linear, exponential, quadratic, or neither. Explain how you know. Linear functions change at a constant rate, so they have a constant
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
. Let’s see if this function
has that by finding the change in y’s and the change in the x’s. +1 +1 +1 +1 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 3 a. The is always − = −3. Since it is constant this is a 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
12
9
-3
-3
6
-3
3
-3
1
1
linear function.
Linear functions change at a constant rate, so they have a constant has that by finding the change in y’s and the change in the x’s. +1 +1 +1 +1 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 2 The is = 2 and b. 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
3
1
9
27
81
1
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
6 1
. Let’s see if this function
= 6 amd
18 1
= 18 and
54 1
= 54.
They are not the same, so this is not a linear function.
+2 +6 +18 +54 Exponential functions multiply by the same thing every time, they have a constant ratio. Let’s see if this function has that by finding what number we multiply each time. The constant ratio is 3 because it multiplies by 3 every time, so this function is exponential. x3
x3
x3
x3
Linear functions change at a constant rate, so they have a constant
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
has that by finding the change in y’s and the change in the x’s. +1 +1 +1 +1 c. 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 −11 The is = −11 and 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
15
4
-1
0
7
1
. Let’s see if this function
−5 1
= −5 amd
1 1
= 1 and
7 1
=7
They are not the same, so this is not a linear function.
+1 +7 -11 -5 Exponential functions multiply by the same thing every time, they have a constant ratio. Let’s see if this function has that by finding what number we multiply each time. 4 To find what we are multiplying by we can divide. = .27, so we must have multiplied by 3.75. 15
We multiplied by different things each time, so this not possible function is not exponential either. x0 x.27 x-.25 Quadratic functions have a constant change of the change. It’s not linear, so the change isn’t constant, but now let’s look at the change of the change. The change of the change is always +6. This is a quadratic function because the change of the change is always constant. +6
+6
+6
Name: ________________________________________________Date: _______________________Hour: ___________ 2. A student takes a subway to a public library. The table shows the distances d (in miles) the student travels in t minutes. Let the time t represent the independent variable. Tell whether the data can be modeled by a linear, exponential, or quadratic function. Explain.
+0.5 1.288
+2
+2
1.57
+.282
The
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
is
0.282 0.5
= 0.564 and
1.14 2
= −7 They are not the
same, so this is not a linear function.
3.85
2.71
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
+1.14
+1.14 We can’t tell if it is exponential or quadratic because the x’s aren’t changing by 1.
3. A store sells custom circular rugs. The table shows the cost c (in dollars) of rugs that have diameters of d feet. Let the diameter d represent the independent variable. Tell whether the data can be modeled by a linear, exponential, or quadratic function. Explain.
74.75
51
32.15
103.4
+23.75 +28.65 Let’s see if it is exponential. Exponential functions multiply the same thing. Let’s see if it is. We find what we are multiplying by dividing. x1
x1
x1 51
32.15
X1.57ish
It is not multiplying the same thing every time, so we know it is not 74.75
103.4
X1.47ish
exponential.
Let’s see if it is quadratic. Quadratic functions have a constant change of the change. +1 +1 +1
32.15
74.75
51
The change of the change is all 4.9, so this is a quadratic function. 103.4
+18,85 +23.75 +28.65 +4.9 +4.9
4. The table shows the breathing rates y (in liters of air per minute) of a cyclist traveling at different speeds x (in miles per hour). Tell whether the data can be modeled by a linear, exponential, or quadratic function. Explain. 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
Linear functions change at a constant rate, so they have a constant
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
. Let’s see if this function
has that by finding the change in y’s and the change in the x’s. +1 +1 +1 It is not adding the same thing every time, so we know it is not linear. 14.2
15.62
18.9002
17.182
+1.42 +1.562 Let’s see if it is exponential. Exponential functions multiply the same thing. Let’s see if it is. We find what we are multiplying by dividing. x1
x1
x1 It multiplies by 1.1 each time, so the function is exponential.
14.2
15.62
x1.1
17.182
x1.1
18.9002
x1.1
Name: ________________________________________________Date: _______________________Hour: ___________ 5. Given the function ℎ(𝑡) = −𝑡 2 + 7𝑡 − 10, for which values of t is the function both positive and increasing? Graph the function on your calculator. The graph is positive when it is above the x-axis and is increasing when you are walking left to right and you are walking uphill. The highlighted area is where it is both positive and increasing. We have to name it with x-values. It happens from x=2 to x=3.5ish.
6. The function 𝑓(𝑡) = −8𝑡 2 + 10𝑡 + 2 gives the height of a ball, in feet, t seconds after it is tossed. What is the average rate of change, in feet per second, over the interval [0.75,1.25]?
The average rate of change is goes on the bottom:
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
. They told us the x-values, that’s what the interval is. X=0.75 and x=1.25. That
. To find our y’s we need to plug the x’s into the equation given.
0.75−1.25
−8(0.75)2 + 10(0.75) + 2 (You can type that in your calculator just like that.) =5 −8(1.25)2 + 10(1.25) + 2 (You can type that in your calculator just like that.) =2 So our y’s are 5 and 2. 5−2 0.75−1.25
=
3 −0.5
= −6 So the rate of change is -6.
7. The graph of a quadratic function 𝑓(𝑥) has a minimum at (-4,-2) and passes through the point (-3,3). The function 𝑔(𝑥) is represented by the equation 𝑔(𝑥) = −(𝑥 − 5)(𝑥 + 3). How much greater is the y-intercept of 𝑓(𝑥) than 𝑔(𝑥)? We need to write the equation for 𝑓(𝑥) and we know it is quadratic, so we are going to use vertex form:𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘. The minimum is the vertex, and the vertex is (h,k). Since our vertex is (-4,-2) our h=-4 and k=-2. Plug that in. 𝑦 = 𝑎(𝑥 − −4)2 + −2 Simplify. 𝑦 = 𝑎(𝑥 + 4)2 − 2 Now we need to find a by plugging in the point they gave us (-3,3) in for x and y and solve. 3 = 𝑎(−3 + 4)2 − 2 Solve. 3 = 𝑎(1)2 − 2 3= 𝑎−2 5 = 𝑎 So a=5, so the equation must be: 𝑦 = 5(𝑥 + 4)2 − 2. Now we can graph both equations (the one we wrote and the one given to us) and see how much higher the y-intercept is on one than the other.
When you graph on your calculator you can see that the y-intercepts are 78 and 15, so the difference between those (78-15) is 63.
Name: ________________________________________________Date: _______________________Hour: ___________ 8. Let x represent the number of years since 1900. The function ℎ(𝑥) = 5𝑥 2 + 5𝑥 + 600 represents the population of Oak Hill. In 1900, Poplar Grove had a population of 100 people. Poplar Grove’s population increased by 4% each year. a. From 1900 to 1950, which town’s population had a greater average rate of change?
We can find the rate of change of Oak Hill because we have an equation. Rate of change is (since we are counting years since 1900.
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
. The x’s are 0 and 50
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 0−50
To find the y’s we need to plug the x’s into the equation we were given. 5(0)2 + 5(0) + 600 this is 0, so the y-value is 0. 5(50)2 + 5(50) + 600 this is 13350, so the y-value is 13350. 0−13350 0−50
=
−13350 −50
= 267 so the average rate of change for Oak Hill is 267.
We can’t find the rate of change of Poplar Grove until we write an equation. We know it is exponential because it has a percent. Exponential percent equations are always 𝑦 = 𝑎(1 + 𝑟) 𝑥 . We start with 100 people so that is our a. We grow by 8%, so .08 is our r. 𝑦 = 100(1 + .08) 𝑥 or 𝑦 = 100(1.08) 𝑥 . Now to find the y’s we need to plug the x’s into the equation we wrote. 100(1.08)0 this is 100, so the y-value is 100 100(1.08)50 this is 4690.16 so the y-value is 4690.16 100−4690.16 0−50
=
−4590.16 −50
= 91.8 so the rate of change of Poplar Grove is about 91.
267 is more than 91, so Oak Hill has a greater rate of change in that time range.
b. Which town will eventually have a greater population?
Exponential always grows faster eventually, so since Poplar Grove is exponential, it will eventually have the highest population.9. In the graph below, function p is an exponential function and function q is a quadratic function. Your friend
says that after about x=3, function q will always have a greater y-value than function p. Is your friend correct? Explain. My friend is not correct. Exponential always grows faster eventually, so since p is exponential, it will eventually have the highest y-values.
10. Explain the mistake in the work below. The linear function continues and so does the quadratic function, so the line and the parabola will cross again. Even though we can’t see it in the picture, we know the system will have two solutions when the line and the parabola cross again later.
Name: ________________________________________________Date: _______________________Hour: ___________ 11. The population of a country is 5 million people and increases by 2% every year. The country’s food supply is sufficient to feed 5 million people and increases at a constant rate that feeds 0.75 million additional people each year. When will the country experience a food shortage?
The population is exponential because it is growing by a percent. Exponential growth is modeled with 𝑦 = 𝑎(1 + 𝑟) 𝑥 .
I start with 5 million people, so 5 is my a (if I am writing the equation in millions) and it is growing by 2%, so 0.2 is r. 𝑦 = 5(1 + .02) 𝑥 or 𝑦 = 5(1.02) 𝑥 . The food is linear because it changes by a constant rate. Linear growth is modeled with 𝑦 = 𝑚𝑥 + 𝑏. I start with food to feed 5 million so my b is 5. I am increasing with 0.75 each year so my m is 0.75. 𝑦 = 0.75𝑥 + 5 Now to find out when there is a shortage we need to find out when they equal each other. We can find that by graphing the two equations and looking where they cross. We have to change the graph window a lot to note where they cross. Notice that my y-values are up to 200 and x-values are up to 500. You can make your calculator show you the intersection by pressing “menu” then “analyze graph”. They cross at (164,128). That means that in 164 years they will have the same amount of people as food, and then in the years after that they won’t have enough food for all of the people.
12. Are the domains for 𝑓(𝑥) = |𝑥 + 2| and 𝑔(𝑥) = |𝑥| + 2 the same or different? Explain. Domain is x-values. If we graph these both on our calculators we can see that in both cases the x-values go on forever in both directions, so the domain is all real numbers.
13. Which of the functions below has the same y-intercept as the function 𝑓(𝑥) = |𝑥 − 2| + 5? Explain.
G(x) has the same y-intercept because it is only multiplying a 3 by the x which smashes the function horizontally but doesn’t change the y-intercept. H(x) is multiplying the 3 outside of the x which stretches the function vertically and could change the y-intercept.
14. Compare the graphs shown below to find what must be the value of k. G(x) is shifted down 3, so k must be -3 because adding a -3 shifts the function down 3. K=-3
Name: ________________________________________________Date: _______________________Hour: ___________ 15. Graph the piecewise function.
The piecewise function is telling us that when our x-values are less than 2 the line will be the line –x. When our x values are bigger than or equal to 2 our line will be x-6. We can type this in our calculator to graph it. The graph will look like this:
