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 Dependent Variable

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# The Intercepts of a Parabola

A y-intercept of a function is a point where the graph crosses the y-axis. That is, a y-intercept is a point where x = 0.

A parabola of the form f(x) = Ax2 + Bx + C always has one y-intercept. To find it, let x = 0 in the equation and simplify:

 Quadratic Equation. Substitute 0 for x. Multiply. Add. f(x) f(0) = Ax2 + Bx + C = A(0)2 + B(0) + C = 0 + 0 + C = C

Thus, the y-intercept of f(x) = Ax2 + Bx + C is the point (0, C).

An x-intercept of a function is a point where the graph crosses the x-axis. That is, a point where y = 0.

A parabola of the form f(x) = Ax2 + Bx + C may have 0, 1, or 2 x-intercepts.

Example 1

Find the y- and x-intercepts of the function: f(x) = x2 - 3x - 10

Solution

The y-intercept is the point (0, C). Thus, it is the point where x = 0.

 Substitute 0 for x. Simplify. f(x) f(0) = x2 - 3x - 10= (0)2 - 3(0) - 10 = -10
So, the y-intercept is (0, -10).

To find the x-intercepts, replace f(x) with 0 and then solve for x.

 Original function. Substitute 0 for f(x). f(x) 0 = x2 - 3x - 10= x2 - 3x - 10

To solve for x:

 Factor. Set each factor equal to 0. Solve each equation. 0 x + 2 x = (x + 2)(x - 5)= 0 or x - 5 = 0 = -2 or x = 5
So, the x-intercepts of f(x) = x2 - 3x - 10 are (-2, 0) and (5, 0).