Question

The tangent line to the graph of \(y=h(x)\) at the point (-1,4) passes through the point (3,6) . Find \(h(-1)\) and \(h^{\prime}(-1)\).

Short answer

The function value at \(-1\) is \(h(-1) = 4\) and the derivative at \(-1\) is \(h^{\prime}(-1) = \frac{1}{2}\).

Step-by-step solution

Identify the given information

Here, we are given that the tangent line to the graph of \(y=h(x)\) at point (-1,4) passes through point (3,6). The task is to find \(h(-1)\) and \(h^{\prime}(-1)\).

Find the slope of the line that passes through the points (-1,4) and (3,6)

The slope of the line through points (x1, y1) and (x2, y2) is given by the formula \[m = \frac{{y2 - y1}}{{x2 - x1}}\]. So, the slope of the tangent line that passes through points (-1,4) and (3,6) is \[m = \frac{{6 - 4}}{{3 - (-1)}} = \frac{1}{2}\].

Find \(h^{\prime}(-1)\)

The slope of the tangent line to the graph at a given point is the same as the derivative of the function at that point. So, \(h^{\prime}(-1) = m = \frac{1}{2}\).

Find \(h(-1)\)

The function value \(h(-1)\) is the y-coordinate of the given point on the graph where the tangent line touches. Hence, \(h(-1)\) is 4.