Lesson 7.2.2
HW: 7-96 to 7-101
Learning Target: Scholars will find linear functions and exponential equations of the form y = abx given
two points.
In this lesson, you will continue your work from Lesson 7.2.1 as you develop a new method to
find linear and exponential equations given two points.
7-92. Mitchell was working on his algebra homework, when suddenly he had an idea about
finding linear equations. He was trying to find the equation of the line that passes through the
points (5, 15) and (3, 7). “Look!” he exclaimed. “We know that the line can be written in the
form y = mx + b, and we also know that the points (5, 15) and (3, 7) have to make the equation
true. So we can substitute in these two points to create a system of equations. When we solve
that, we’ll know the values of m and b, and we’ll have our equation!”
1. What is Mitchell talking about? Use his method to find the equation of the line
through the points (5, 15) and (3, 7).
2. Will Mitchell's method work to find the equation of a line through any two
points? Justify your answer.
7-93. Use Mitchell’s method from problem 7-92 to find the equation of the line that passes
through the points (2, 3) and (5, –6).
7-94. Can Mitchell’s method from problem 7-92 be used to find the exponential function that
passes through the points (2, 16) and (6, 256)? Consider this as you answer the questions below.
3. What is the general form for an exponential function that has an asymptote at y =
0?
4. Use the two points that you know to create a system of equations.
5. Solve both equations for a. Then use the Equal Values Method to solve your
system of equations for b. Find a, and write the equation that goes through the
two points.
7-95. Find an exponential function that passes through each pair of points.
6. (−1, −2) and (3, −162)
7. (2, 1.75) and (−2, 28)
7-96. Find an exponential function that passes through each pair of points.
1. (1, 7.5) and (3, 16.875)
2. (−1, 1.25) and (3, 0.032)
7-97. Consider the pattern at right.
3. Continue the pattern to find
4. What is the value of
, and
.
?
5. Write a conjecture about how to rewrite
without a negative exponent.
7-98. Find the domain and range for each of the relations graphed below.
a.
b.
d.
c.
7-99. If f(x) = 3(2)x , find the value of the expressions in parts (a) through (c) below. Then
complete parts (d) through (f). 7-99 HW eTool (Desmos).
6. f(−1)
7. f(0)
8. f(1)
9. What value of x gives f(x) = 12?
10. Where does the graph of this function cross the x-axis? The y-axis?
11. If
, find f(x) · g(x).
7-100. Show two steps to simplify each of the following expressions, and then calculate the
value of each expression.
12. 642/3
13. 255/2
14. 817/4
7-101. Copy and complete each of the Diamond Problems below. The pattern used in the
Diamond Problems is shown at right.
Lesson 7.2.2








7-92. See below:
1. The system: 15 = 5m + b and 7 = 3m + b; m = 4 and b = –5, so y = 4x – 5.
2. Not quite; his method will work for all linear equations except vertical lines.
7-93. y = –3x + 9
7-94. See below:
1. y = abx
2. 16 = ab2 and 256 = ab6
3. b = 2 and a = 4, so y = 4 · 2x
7-95. See below:
1. y = –6 · 3x
2. y = 7(0.5)x
7-96. See below:
1. y = 5 · 1.5x
2. y = 0.5(0.4)x
7-97. See below:
1. 2, 4, 8, 16
2. 2n
3. an
7-98. See below:
1. x = 0, 1 2 and y = –2, 0, 1
2. –1 ≤ x ≤ 1 and –1 ≤ y ≤ 2
3. x ≤ 2 and y ≥ –2
4. x: all real numbers and y ≥ –1
7-99. See below:
1.
2.
3.
4.
5.

3
6
2
Never; (0, 3)
6.
7-100. See below:
1. 16
2. 3125
3. 2187

7-101. See the answers in bold in the diamonds below: