Which point is an $x$-intercept of the quadratic function $f(x)=(x-8)(x+9)$ ?
To identify the x-intercepts of the quadratic function \(f(x) = (x-8)(x+9)\), we first recall that an x-intercept occurs at any point \((x, y)\) where the function value is zero. That is, we seek values of \(x\) such that \(f(x) = 0\). Substituting the given expression for \(f(x)\), we set up the equation:
\[(x-8)(x+9) = 0\]
According to the zero-product property, a product of factors is equal to zero if and only if at least one of the factors is equal to zero. Therefore, we can set each factor to zero to solve for \(x\):
\[x - 8 = 0 \implies x = 8\]
\[x + 9 = 0 \implies x = -9\]
These are the x-coordinates of the x-intercepts. Since x-intercepts are points on the coordinate plane where the y-coordinate is \(0\), the corresponding points are \((8, 0)\) and \((-9, 0)\).
We compare our findings with the given options:
We can see that point \((-9, 0)\) matches option (c). Thus, \((-9, 0)\) is an x-intercept of the function \(f(x) = (x-8)(x+9)\).
X-intercepts are crucial for understanding the behavior of physical systems modeled by quadratic equations. For instance, if you are calculating the trajectory of a projectile (like a ball thrown in the air), the x-intercepts represent the ground level, identifying when the object starts or finishes its flight. By solving \(f(x) = 0\), engineers and scientists can predict exactly when an object will hit the ground, helping in safety design and performance optimization in sports or aerospace engineering.