If the graph of a line has a positive slope and a negative y-intercept, what happens to the x-intercept if the slope and y-intercept are doubled?
A) The x-intercept becomes 4 times larger
B) The x-intercept becomes twice as large
C) The x-intercept becomes 1/4th as large
D) The x-intercept remains the same
I need the answer, but also how I would figure this out myself. Thanks!
d) stays the same
To determine how doubling the slope and y-intercept of a line affects the x-intercept, we can use the equation of a line in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
Let's assume the original line has a slope of m and a y-intercept of b. Therefore, its equation is y = mx + b.
When doubling the slope and y-intercept, the new equation becomes y = 2m(x) + 2b.
To find the x-intercept, we set y = 0 in the equation and solve for x: 0 = 2m(x) + 2b.
Dividing both sides of the equation by 2m gives us 0 = x + b/m.
Subtracting b/m from both sides gives us -b/m = x.
Comparing this result to the original equation y = mx + b, we can see that the x-intercept has changed to -b/m.
Since the slope and y-intercept were doubled, the new x-intercept is now -2b/(2m) = -b/m.
From this analysis, we can see that doubling both the slope and y-intercept does not change the x-intercept. Therefore, the answer is D) The x-intercept remains the same.
To determine what happens to the x-intercept when the slope and y-intercept of a line are doubled, we can use the equation of a line: y = mx + b, where m is the slope, b is the y-intercept, and (x, y) represents any point on the line.
Given that the original line has a positive slope and a negative y-intercept, we can represent it as y = mx - b (with the negative sign in front of the y-intercept).
Let's assume the x-intercept of the original line is represented by the point (x1, 0). Since the x-intercept is the point where the line crosses the x-axis, the y-coordinate at the x-intercept should be 0.
Substituting these values into the equation, we have:
0 = mx1 - b
To find the x-intercept, we solve for x1:
mx1 = b
x1 = b/m
Now, if we double both the slope and the y-intercept, the new equation of the line becomes y = 2m2x - 2b.
We can find the new x-intercept by setting y equal to 0:
0 = 2m2x - 2b
Dividing both sides by 2m2, we get:
0 = x - b/m
Now, we can compare this to the original x-intercept equation:
x1 = b/m
From the comparison, we can see that the new x-intercept, x2, is equal to the original x-intercept, x1.
Therefore, the x-intercept remains the same when the slope and y-intercept of a line are doubled.
So the correct answer is D) The x-intercept remains the same.