Consider the quadratic function \( f(x)=x^{2}+2 x-15 \). Determine the following: The smallest \( x \)-intercept is \( x=\square \) The largest \( x \)-intercept is \( x=\square \) The \( y \)-intercept is \( y=\square \)

To find the \( x \)-intercepts of the quadratic function \( f(x) = x^2 + 2x - 15 \), we need to set \( f(x) = 0 \) and solve for \( x \): \[ x^2 + 2x - 15 = 0 \] We can factor this quadratic equation. We are looking for two numbers that multiply to \(-15\) (the constant term) and add to \(2\) (the coefficient of \(x\)). The numbers \(5\) and \(-3\) satisfy these conditions: \[ (x + 5)(x - 3) = 0 \] Setting each factor to zero gives us the \( x \)-intercepts: \[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \] \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] Thus, the smallest \( x \)-intercept is \( x = -5 \) and the largest \( x \)-intercept is \( x = 3 \). Next, we find the \( y \)-intercept by evaluating \( f(0) \): \[ f(0) = 0^2 + 2(0) - 15 = -15 \] Therefore, the \( y \)-intercept is \( y = -15 \). In summary, we have: - The smallest \( x \)-intercept is \( x = -5 \) - The largest \( x \)-intercept is \( x = 3 \) - The \( y \)-intercept is \( y = -15 \) So, the final answers are: The smallest \( x \)-intercept is \( x = -5 \) The largest \( x \)-intercept is \( x = 3 \) The \( y \)-intercept is \( y = -15 \)