Write the slope-intercept equation of the function whose graph satisfies the given conditions. The graph of f passes through (-9,9) and is perpendicular to the line that has an x-intercept of 4 and a y-intercept of -12.
First, we need to find the slope of the line that has an x-intercept of 4 and a y-intercept of -12. We can start by setting up the equation of the line using the intercepts:
y = mx + b
where m is the slope and b is the y-intercept. We know that the y-intercept is -12, so we can plug that in:
y = mx - 12
To find the slope, we can use the fact that the line passes through the two intercepts. The x-intercept is (4, 0), so the point-slope form of the equation would be:
y - 0 = m(x - 4)
Simplifying this gives:
y = mx - 4m
Now we can set this equal to the y-intercept and solve for the slope:
-12 = 4m
m = -3
So the slope of the line is -3. Since we want the line perpendicular to this one, we know that the slope of our function must be the negative reciprocal of -3, which is 1/3.
Now we can use the point-slope form of the equation to find the equation of our function, using the point (-9, 9):
y - 9 = (1/3)(x + 9)
Simplifying this gives:
y = (1/3)x + 12
So the slope-intercept equation of the function is y = (1/3)x + 12.
To find the slope-intercept equation of the function, we need to determine the slope of the line that is perpendicular to the given line.
The given line has an x-intercept of 4 and a y-intercept of -12, so we can write its equation in slope-intercept form as:
y = mx + b, where m represents the slope and b represents the y-intercept.
The y-intercept is -12, so b = -12.
To find the slope, we need to calculate the difference in y-coordinates divided by the difference in x-coordinates between the two intercept points:
m = (y2 - y1) / (x2 - x1)
The x-intercept is 4, so (x1, y1) = (4, 0).
The y-intercept is -12, so (x2, y2) = (0, -12).
m = (-12 - 0) / (0 - 4) = -12 / -4 = 3
Since the line we're looking for is perpendicular to this line, the slope of the new line is the negative reciprocal of 3.
The negative reciprocal of 3 is -1/3.
Now that we have the slope and the point (-9, 9) that the graph of f passes through, we can substitute these values into the point-slope form of a linear equation:
y - y1 = m(x - x1)
Using (-9, 9) and m = -1/3, we have:
y - 9 = (-1/3)(x - (-9))
y - 9 = (-1/3)(x + 9)
y - 9 = (-1/3)x - 3
y = (-1/3)x + 6
Therefore, the slope-intercept equation of the function that satisfies the given conditions is y = (-1/3)x + 6.