Verified Answer :

  • Set the function \(f(x) = (x - 8)(x + 9)\) to zero to find the x-intercepts.
  • Apply the zero product property to solve \(x - 8 = 0\) and \(x + 9 = 0\).
  • Find the x-intercepts as \((8, 0)\) and \((-9, 0)\).
  • Identify that \((-9, 0)\) matches option c. The final answer is \(\boxed{c}\).

Explanation

  1. Understanding x-intercepts

The x-intercepts of a function \(f(x)\) are the points \((x, 0)\) where the function's value is zero. For the given quadratic function \(f(x) = (x - 8)(x + 9)\), finding the x-intercepts involves solving the equation \(f(x) = 0\), which is:
\[(x - 8)(x + 9) = 0\]

  1. Solving the equation for x-intercepts

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. Applying this property to our equation:
\[x - 8 = 0 \quad \text{or} \quad x + 9 = 0\]
Solving each linear equation individually:

  1. For \(x - 8 = 0\), we add \(8\) to both sides to get \(x = 8\).
  2. For \(x + 9 = 0\), we subtract \(9\) from both sides to get \(x = -9\).

Thus, the x-coordinates of the intercepts are \(x = 8\) and \(x = -9\). The corresponding points are \((8, 0)\) and \((-9, 0)\).

  1. Identifying the correct option

Comparing our results \((8, 0)\) and \((-9, 0)\) to the provided options:

A\((9, 0)\)
B\((0, -8)\)
C\((-9, 0)\)

We can see that point \((-9, 0)\) matches option c. Therefore, \((-9, 0)\) is an x-intercept of the function.

Examples

In real-world applications, finding the x-intercepts of a quadratic function helps us determine where a projectile lands or when an object returns to ground level. For example, if the height of a tossed ball is modeled by a quadratic function, setting the height to zero and solving for time gives us the moments the ball hits the ground. Understanding these roots is essential for analyzing physical motion and optimization problems in fields like engineering and physics.

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