Calculating x and y intercepts

An SAT Math micro-topic under Graphs of linear equations and functions (Algebra). Free to read — no account needed.

The intercepts of a graph are the points where it meets the axes. The y-intercept is where the graph crosses the y-axis; every point on the y-axis has x=0x = 0. The x-intercept is where it crosses the x-axis; every point on the x-axis has y=0y = 0.
intercepts_line.png
In the graph above, the line y=−2x+4y = -2x + 4 crosses the y-axis at (0,4)(0, 4) and the x-axis at (2,0)(2, 0).
To find the y-intercept, set x=0x = 0: y=−2(0)+4=4y = -2(0) + 4 = 4, giving (0,4)(0, 4).
To find the x-intercept, set y=0y = 0: 0=−2x+40 = -2x + 4, so x=2x = 2, giving (2,0)(2, 0).
The same idea works for any function, not just lines. For a curve like f(x)=a(2x)+cf(x) = a(2^x) + c, the y-intercept is f(0)=a+cf(0) = a + c, because 20=12^0 = 1.

Worked examples

Find the y-intercept of y=3x−6y = 3x - 6.
Set x=0x = 0: y=3(0)−6=−6y = 3(0) - 6 = -6.
So the y-intercept is (0,−6)(0, -6).
Find the x-intercept of y=3x−6y = 3x - 6.
Set y=0y = 0: 0=3x−60 = 3x - 6, so 3x=63x = 6 and x=2x = 2.
So the x-intercept is (2,0)(2, 0).
Find the y-intercept of f(x)=5(2x)−1f(x) = 5(2^x) - 1.
Set x=0x = 0: f(0)=5(20)−1=5(1)−1=4f(0) = 5(2^0) - 1 = 5(1) - 1 = 4, since 20=12^0 = 1.
So the y-intercept is (0,4)(0, 4).

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