Given the function g(x) = 8x − 2, compare and contrast g(−2) and g(4). Choose the statement that is true concerning these two values.
a. The value of g(−2) is larger than the value of g(4).
b. The value of g(−2) is the same as the value of g(4).
c. The value of g(−2) is smaller than the value of g(4).
d. The values of g(−2) and g(4) cannot be compared.
To compare g(-2) and g(4), we substitute -2 into the function and substitute 4 into the function and see which value is greater.
g(-2) = 8(-2) - 2 = -16 - 2 = -18
g(4) = 8(4) - 2 = 32 - 2 = 30
Since -18 is less than 30, we can conclude that the value of g(-2) is smaller than the value of g(4).
Therefore, the correct statement is:
c. The value of g(−2) is smaller than the value of g(4).
If h(x) = −2x − 10, find h(−4).
a. −2
b. −18
c. −3
d. −16
To find h(-4), we substitute -4 into the function:
h(-4) = -2(-4) - 10 = 8 - 10 = -2
Therefore, h(-4) = -2.
The correct statement is:
a. -2
Write the equation of the line that passes through the points (3, 6) and (5, 18) using function notation.
a. f(x) = 2x + 12
b. f(x) = 6x − 12
c. y = 2x + 12
d. y = 6x − 12
To find the equation of the line that passes through the points (3, 6) and (5, 18), we can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.
First, let's find the slope (m):
m = (change in y) / (change in x)
m = (18 - 6) / (5 - 3)
= 12 / 2
= 6
Now that we have the slope, we can substitute the slope and one of the given points into the slope-intercept form to find the y-intercept (b).
6 = 6(3) + b
6 = 18 + b
b = 6 - 18
b = -12
So the y-intercept is -12.
Therefore, the equation of the line that passes through the points (3, 6) and (5, 18) is:
y = 6x - 12.
In function notation, this can be written as:
f(x) = 6x - 12.
Therefore, the correct answer is:
b. f(x) = 6x - 12.
The graph shows the distance (y) between Hannah and her home, in km, after certain amounts of time (x), in hours.
Graph of first line going through ordered pairs 0, 4 and 2, 4. Graph of second line going through ordered pairs 2, 4 and 3, 2. Graph of third line going through ordered pairs 3, 2 and 4, 4. Graph of fourth line going through ordered pairs 4, 4 and 6, 4.
Which of the following statements best describes Hannah's distance from her home?
It is decreasing in the interval 2 < x < 4 hours.
It is increasing in the interval 2 < x < 3 hours.
It is decreasing in the interval 2 < x < 3 hours.
It is increasing in the interval 2 < x < 4 hours.
To determine the statement that best describes Hannah's distance from her home, we need to examine the intervals on the x-axis where the distance is increasing or decreasing based on the given graph.
From the graph, we can see that the first line segment (0, 4) to (2, 4) has a horizontal line, indicating that the distance remains constant at 4 km in the interval 0 < x < 2 hours.
The second line segment (2, 4) to (3, 2) has a negative slope, indicating that the distance is decreasing in the interval 2 < x < 3 hours.
The third line segment (3, 2) to (4, 4) has a positive slope, indicating that the distance is increasing in the interval 3 < x < 4 hours.
The fourth line segment (4, 4) to (6, 4) has a horizontal line, indicating that the distance remains constant at 4 km in the interval 4 < x < 6 hours.
Therefore, the correct statement that best describes Hannah's distance from her home is:
It is decreasing in the interval 2 < x < 3 hours.
The answer is:
c. It is decreasing in the interval 2 < x < 3 hours.