two linear functions are described below
Function f(x) has the equation f (x)=3x-4
function g(x) able of values shown below.
0 4
3 5
6 6
9 7
Which statement is true regarding the functions and ?
A.
The slopes of the two functions are the same.
B.
The slopes of the two functions are opposites.
C.
The y-intercepts of the two functions are the same.
D.
The y-intercepts of the two functions are opposites.
The equation for function f(x) is f(x) = 3x - 4.
To find the slope of function f(x), we can observe that the coefficient of x is 3. Therefore, the slope of function f(x) is 3.
To find the slope of function g(x), we can observe the change in the y-values as x increases by 3 units. We see that for every increase of 3 units in x, the corresponding y-value increases by 1 unit. Therefore, the slope of function g(x) is 1/3.
Based on the calculated slopes, we can conclude that statement A ("The slopes of the two functions are the same") is false.
To find the y-intercept of function f(x), we can set x = 0 in the equation f(x) = 3x - 4. When x = 0, f(x) = -4. Therefore, the y-intercept of function f(x) is -4.
To find the y-intercept of function g(x), we can observe the value of y when x = 0. We see that when x = 0, y = 4. Therefore, the y-intercept of function g(x) is 4.
Based on the calculated y-intercepts, we can conclude that statement C ("The y-intercepts of the two functions are the same") is false.
Therefore, the correct statement regarding the functions f(x) and g(x) is:
D. The y-intercepts of the two functions are opposites.
To determine which statement is true regarding the functions f(x) and g(x), let's analyze the given information.
The equation for f(x) is f(x) = 3x - 4. This represents a linear function with a slope of 3 and a y-intercept of -4.
Now let's look at the values of g(x):
0 4
3 5
6 6
9 7
By analyzing these points, we can calculate the slope of g(x):
Slope = (change in y) / (change in x)
Slope from (0, 4) to (3, 5) = (5 - 4) / (3 - 0) = 1 / 3
Slope from (3, 5) to (6, 6) = (6 - 5) / (6 - 3) = 1 / 3
Slope from (6, 6) to (9, 7) = (7 - 6) / (9 - 6) = 1 / 3
The slopes of g(x) are all equal to 1/3.
Comparing the slopes of f(x) and g(x), we find that they are not the same. The slope of f(x) is 3, whereas the slope of g(x) is 1/3.
Therefore, the correct statement is:
A. The slopes of the two functions are NOT the same.
To compare the slopes of the two functions, we'll first need to determine the slope of function f(x). The equation of function f(x) is given as f(x) = 3x - 4.
The slope-intercept form of a linear function is y = mx + b, where m is the slope and b is the y-intercept. By comparing the equation of f(x) to the slope-intercept form, we can see that the slope of f(x) is 3.
Now, let's examine the table given for function g(x). We see that as the x-values increase by 3 (from 0 to 3, 3 to 6, and 6 to 9), the corresponding y-values increase by 1 (from 4 to 5, 5 to 6, and 6 to 7).
If we calculate the difference in y-coordinates divided by the difference in x-coordinates for any two points on the table, we will get the slope of function g(x).
For example, if we choose the first two points (0, 4) and (3, 5), the difference in y-coordinates is 5 - 4 = 1, and the difference in x-coordinates is 3 - 0 = 3. Therefore, the slope of g(x) for these points is 1/3.
By repeating this calculation for the other pairs of points, we find that all the slopes of g(x) are equal to 1/3.
Comparing the slopes of f(x) and g(x), we see that the slopes of the two functions are indeed the same. Therefore, the correct statement is:
A. The slopes of the two functions are the same.