To find the intercepts of a quadratic function, you need to find the points where the parabola intersects the x-axis (x-intercepts) and the y-axis (y-intercept). Here's how:
Finding the x-intercepts (also known as roots or zeros)
The x-intercepts are the points where the quadratic function, f(x) or y, equals zero.
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Set the quadratic function equal to zero: Replace f(x) or y with 0. This gives you an equation in the form ax² + bx + c = 0.
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Solve for x: There are several ways to solve this quadratic equation:
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Factoring: If the quadratic expression can be factored easily, factor it and set each factor equal to zero. For example, if the equation is x² - 5x + 6 = 0, it factors to (x - 2)(x - 3) = 0. Setting each factor to zero gives x - 2 = 0 and x - 3 = 0, so x = 2 and x = 3.
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Quadratic Formula: Use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. This formula works for any quadratic equation.
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Completing the Square: This method involves manipulating the equation to create a perfect square trinomial.
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Write the x-intercepts as ordered pairs: The solutions for x are the x-coordinates of the x-intercepts. The y-coordinate is always 0 at the x-intercept. So, if you find that x = x₁ and x = x₂, the x-intercepts are (x₁, 0) and (x₂, 0).
Finding the y-intercept
The y-intercept is the point where the quadratic function intersects the y-axis. This occurs when x = 0.
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Substitute x = 0 into the quadratic function: Replace every x in the equation with 0. For example, if f(x) = 2x² + 3x - 4, then f(0) = 2(0)² + 3(0) - 4 = -4.
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Write the y-intercept as an ordered pair: The y-intercept is the point (0, f(0)) or (0, y). In the previous example, the y-intercept is (0, -4).
Example
Consider the quadratic function f(x) = x² - 4x + 3.
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X-intercepts:
- Set f(x) = 0: x² - 4x + 3 = 0
- Factor: (x - 1)(x - 3) = 0
- Solve for x: x = 1 and x = 3
- X-intercepts: (1, 0) and (3, 0)
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Y-intercept:
- Set x = 0: f(0) = (0)² - 4(0) + 3 = 3
- Y-intercept: (0, 3)
