Find a linear function \( h \), given \( h(3)=-7 \) and \( h(-1)=13 \). Then find \( h(4) \). \( h(x)=\square \) (Type an expression using \( x \) as the variable. Simplify your answer.)

To find the linear function \( h(x) \) that satisfies the given conditions \( h(3) = -7 \) and \( h(-1) = 13 \), follow these steps: 1. **Assume the form of the linear function:** \[ h(x) = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. 2. **Set up equations using the given points:** \[ \begin{cases} 3m + b = -7 \quad \text{(from } h(3) = -7\text{)} \\ -m + b = 13 \quad \text{(from } h(-1) = 13\text{)} \end{cases} \] 3. **Solve the system of equations:** - Subtract the second equation from the first: \[ (3m + b) - (-m + b) = -7 - 13 \\ 4m = -20 \\ m = -5 \] - Substitute \( m = -5 \) into the second equation to find \( b \): \[ -(-5) + b = 13 \\ 5 + b = 13 \\ b = 8 \] 4. **Write the linear function:** \[ h(x) = -5x + 8 \] 5. **Find \( h(4) \):** \[ h(4) = -5(4) + 8 = -20 + 8 = -12 \] **Final Answer:** \[ h(x) = -5x + 8 \] \[ h(4) = -12 \]