The table below represents a linear function f(x) and the equation represents a function g(x):
x f(x)
−1 −5
0 −1
1 3
g(x)
g(x) = 2x − 7
Part A: Write a sentence to compare the slope of the two functions and show the steps you used to determine the slope? (6 points)
Part B: Which function has a greater y-intercept? (4 points)
Part A: To compare the slopes of the two functions, we need to calculate the slope of each function. The slope of a linear function can be found using the formula:
slope = (change in y)/(change in x)
For the linear function f(x), we have the following points:
(-1, -5), (0, -1), (1, 3)
Let's calculate the slope:
Slope of f(x) = (change in y) / (change in x)
For the point (-1, -5) to (0, -1):
Change in y = -1 - (-5) = 4
Change in x = 0 - (-1) = 1
Slope of f(x) = 4/1 = 4
For the point (0, -1) to (1, 3):
Change in y = 3 - (-1) = 4
Change in x = 1 - 0 = 1
Slope of f(x) = 4/1 = 4
So the slope of the function f(x) is 4.
Now let's find the slope of the function g(x):
g(x) = 2x - 7
We can see that the coefficient of x is 2, which means the slope of g(x) is 2.
Comparing the slopes of the two functions:
The slope of f(x) is 4, while the slope of g(x) is 2.
Therefore, the slope of f(x) is greater than the slope of g(x).
Part B: To determine which function has a greater y-intercept, we need to compare the y-intercepts of both functions. The y-intercept of a linear function occurs when x = 0, so we can substitute x = 0 into the equations of f(x) and g(x) to find their respective y-intercepts.
For f(x), when x = 0:
f(0) = -1
So the y-intercept of f(x) is -1.
For g(x), when x = 0:
g(0) = 2(0) - 7 = -7
So the y-intercept of g(x) is -7.
Comparing the y-intercepts:
The y-intercept of f(x) is -1, while the y-intercept of g(x) is -7.
Therefore, the function f(x) has a greater y-intercept than g(x).
Part A: To compare the slope of the two functions, we need to determine the slope of each function. The slope of a linear function can be found by calculating the change in y divided by the change in x.
For function f(x), using the points (-1, -5) and (1, 3), we can calculate the slope as follows:
slope = (change in y) / (change in x)
= (3 - (-5)) / (1 - (-1))
= 8 / 2
= 4
For function g(x), the equation is given as g(x) = 2x - 7. Since this is in slope-intercept form (y = mx + b), we know that the slope (m) is equal to 2.
Therefore, the slope of function f(x) is 4, and the slope of function g(x) is 2.
Part B: To determine which function has a greater y-intercept, we need to compare the values of b in each function's equation.
For function f(x), we can find the y-intercept by substituting x as 0 into the function:
f(0) = -1
For function g(x), we can find the y-intercept by setting x as 0 in the equation and solving for y:
g(0) = 2(0) - 7
= -7
Comparing the y-intercepts, we see that the y-intercept of function f(x) is -1, and the y-intercept of function g(x) is -7.
Therefore, the function with the greater y-intercept is f(x).
g(x) is written in the slope-intercept form. Its slope and y-intercept should be obvious.
For f(x), note that
y changes by 2 when x changes by 1
f(0) = -1
...