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Determine whether each table represents a linear or nonlinear function. Explain.
SOLUTION: Determine how the values change for x and y. As x increases by 2 each time, y increases by 1 each time. The rate of change is constant, so this function is linear. ANSWER: Linear; rate of change is constant; as x increases by 2, y increases by 1.
SOLUTION: Determine how the values change for x and y. Although x increases by 1 each time, y increases by a larger amount each time. The rate of change is not constant, so this function is nonlinear. ANSWER: Nonlinear; rate of change is not constant.
SOLUTION:
Determine how the values change for x and y. As x increases by 5 each time, y increases by 15 each time. The rate of change is constant, so this function is linear. ANSWER: Linear; rate of change is constant; as x increases by 5, y increases by 15.
SOLUTION: Determine how the values change for x and y. Although x increases by 2 each time, y decreases by a larger amount each time. The rate of change is not constant, so this function is nonlinear. ANSWER: Nonlinear; rate of change is not constant.
- The table shows the height of several buildings in Chicago. Use the table to determine whether the height of the building is a linear function of the number of stories. Explain. SOLUTION: Determine how the values change for x and y. Although x increases by 5 each time, y changes by different amounts each time. The rate of change is not constant, so this function is nonlinear. ANSWER: No; the rate of change is not constant.
- There are 3,600 seconds in one hour. The total seconds is a function of the hours. Does this situation represent a linear or nonlinear function? Explain. SOLUTION: Make a table with two rows and five columns. Label the first row Hours( h ) and the second row Seconds( s ). Choose values for h and evaluate them in the expression z = 3,600 h. Hours ( h ) 1 2 3 4 Seconds ( s ) 3,600 7,200 10,800 14, Graph the points ( h , s ) on a coordinate plane. The ordered pairs (hours, seconds) lie on a straight line, so the situation represents a linear function. ANSWER: Linear; sample answer: If you graph the function, the ordered pairs (hours, seconds) lie on a straight line.
Linear; sample answer: The points lie on a straight line. ANSWER: Linear; sample answer: The points lie on a straight line.
- y = SOLUTION: Select any four values for the domain, x. Substitute these values for x to find the value of y. Graph each ordered pair. Draw a line that passes through each point. Sample Table: x y ( x , y )
0 Undefined Undefined 3 6 Graph: Nonlinear; sample answer: The graph is a curve. ANSWER: Nonlinear; sample answer: The graph is a curve.
- y =
SOLUTION:
Linear functions have graphs that are straight lines. The first graph is the only graph that is a straight line. The rest are nonlinear. ANSWER:
- Persevere with Problems Does the graph shown represent a linear function? Explain. SOLUTION: The graph is a line, however, the graphs of vertical lines are not functions because there is more than one value of y that corresponds to x = 2. So, the graph does not represent a linear function. ANSWER: No; sample answer: the graphs of vertical lines are not functions because there is more than one value of y that corresponds to x = 2.
- Model with Mathematics Give an example of a situation that can be represented by a nonlinear function. SOLUTION:
Sample answer: Every hour the number of bacteria in a petri dish doubles.
ANSWER:
Sample answer: Every hour the number of bacteria in a petri dish doubles.
SOLUTION:
Determine how the values change for x and y. As x increases by 4 each time, y decreases by 3 each time. The rate of change is constant, so this function is linear. ANSWER: Linear; rate of change is constant; as x increases by 4, y decreases by 3.
- Copy and Solve The area of a square is a function of its perimeter. Graph the function on a separate sheet of grid paper. Explain whether the function is linear and if the graph is increasing or decreasing. SOLUTION: Create a table of values that has two rows and five columns. Label the rows for perimeter P and area A. Choose a side length for each square and then calculate the perimeter and area to complete the table. Use the formulas P= 4 s
and A = s^2 , where P is the perimeter of a square, A is the area of a square, and s is the side length of the square.
Perimeter P (units) 0 4 8 16 Area A (sq. units) 0 1 4 16 Graph the points as ( P , A ) on a coordinate plane.
Sample answer: The function is nonlinear because the graph of the function is not a straight line and the line is increasing.. ANSWER: Sample answer: The function is nonlinear because the graph of the function is not a straight line and the line is increasing.
- Multiple Representations Recall that the circumference of a circle is equal to two times Ď times its radius and that the area of a circle is equal to Ď times the square of the radius. a. Tables Complete the table showing the circumference and area of circles with radius r.
b. Labe the x -axis as radius. Label the y -axis as circumference/area. Graph the ordered pairs (radius, circumference) then connect those points. Then graph the ordered pairs (radius, area) and connect those points.
c. When you connect the points for the circumference, the graph is straight line so the function is linear. When you
connect the points for area, they form a curve so the function is nonlinear. ANSWER: a. b.
c. Circumference: linear; sample answer: When the ordered pairs are graphed, the points fall in a line. Area: nonlinear; sample answer: When the ordered pairs are graphed, the points do not fall in a line.
20. Determine if each table represents a linear or nonlinear function.
SOLUTION:
This table represents a linear function because the x -values increase by 2.5 each time and the y -values increase by 0.75 each time.
true or false. a. The function representing Jungâs savings is nonlinear. b. The function representing Miguelâs savings is linear. c. After 1 year, Jung will have saved $320. SOLUTION: Create a table with three columns, one to represent the number of months, m , one for Jungâs total savings, and one for Miguelâs total savings. Jung starts with $200 and increases his savings by $10 a month. Miguel starts with $200 but he earns 10% interest on his money each month.
Compare the rates of change. Jungâs savings represent a linear function because the rate of change is constant. As
the months increase by 1, the savings increase by 10. Miguelâs savings represent a nonlinear function because as the months increase by 1, the total savings increases at a growing rate.
Jung's account can be represented by using slope-intercept form, y = 10 x + 200 , where x represents the number
of months and y represents the amount in the savings account. Since there are 12 months in a year, the amount in
Jung's account after one year can be found by replacing x with 12.
y = 10(12) + 200
After one year there will be $320 in Jung's account.
ANSWER:
a. False
b. False
c. True
Find each function value.
- f (5) if f ( x ) = 3 x + 4 SOLUTION: Substitute 5 for x into the function rule and evaluate. ANSWER: 19
- f (â3) if f ( x ) = 2 x â 8 SOLUTION: Substitute â3 for x into the function rule and evaluate. ANSWER:
- f (7) if f ( x ) = 9 x â 24 SOLUTION: Substitute 7 for x into the function rule and evaluate. ANSWER: 39
