Find the slope of the function's graph at the given point. \( f(x)=\sqrt{3 x+7},(3,4) \) The slope of the function's graph at \( (3,4) \) is \( \frac{3}{8} \). (Simplify your answer.) The equation for the tangent line through \( (3,4) \) is \( y=\square \)

To find the slope of the function's graph at the given point \( (3,4) \), we need to calculate the derivative of the function \( f(x) = \sqrt{3x + 7} \) and then evaluate it at \( x = 3 \). ### Step 1: Find the derivative of \( f(x) \) The function can be rewritten as: \[ f(x) = (3x + 7)^{1/2} \] Using the power rule and the chain rule, the derivative \( f'(x) \) is given by: \[ f'(x) = \frac{1}{2}(3x + 7)^{-1/2} \cdot 3 \] ### Step 2: Simplify the derivative Now, let's simplify \( f'(x) \): \[ f'(x) = \frac{3}{2\sqrt{3x + 7}} \] ### Step 3: Evaluate the derivative at \( x = 3 \) Now we substitute \( x = 3 \) into the derivative: \[ f'(3) = \frac{3}{2\sqrt{3(3) + 7}} = \frac{3}{2\sqrt{9 + 7}} = \frac{3}{2\sqrt{16}} = \frac{3}{2 \cdot 4} = \frac{3}{8} \] So, the slope of the function's graph at the point \( (3,4) \) is indeed \( \frac{3}{8} \). ### Step 4: Find the equation of the tangent line The equation of the tangent line can be expressed in point-slope form: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope, and \( (x_1, y_1) \) is the point of tangency \( (3, 4) \). Substituting the values: \[ y - 4 = \frac{3}{8}(x - 3) \] ### Step 5: Simplify the equation Now, let's simplify this equation: \[ y - 4 = \frac{3}{8}x - \frac{3}{8} \cdot 3 \] \[ y - 4 = \frac{3}{8}x - \frac{9}{8} \] \[ y = \frac{3}{8}x - \frac{9}{8} + 4 \] \[ y = \frac{3}{8}x - \frac{9}{8} + \frac{32}{8} \] \[ y = \frac{3}{8}x + \frac{23}{8} \] Thus, the equation for the tangent line through \( (3,4) \) is: \[ y = \frac{3}{8}x + \frac{23}{8} \] ### Final Answer The slope of the function's graph at \( (3,4) \) is \( \frac{3}{8} \), and the equation for the tangent line through \( (3,4) \) is: \[ y = \frac{3}{8}x + \frac{23}{8} \]