Question

8 mark for review graph of f the graph of f, the derivative of a function f, is shown above. the points (2,6) and (4,18) are on the graph of f. which of the following is an equation of the line tangent to the graph of f at x = 2?

Explanation:

Step1: Recall the tangent line formula

The equation of the tangent line to the graph of \(y = f(x)\) at \(x = a\) is \(y - f(a)=f^{\prime}(a)(x - a)\).

Step2: Identify \(a\), \(f(a)\) and \(f^{\prime}(a)\)

Here \(a = 2\). Given that the point \((2,6)\) is on the graph of \(f\), so \(f(2)=6\). Also, \(f^{\prime}(2)\) is the value of the derivative at \(x = 2\). From the graph of \(y = f^{\prime}(x)\), when \(x = 2\), \(y=f^{\prime}(2)=5\) (by looking at the \(y -\) value of the graph of \(y = f^{\prime}(x)\) at \(x = 2\)).

Step3: Substitute into the tangent line formula

Substitute \(a = 2\), \(f(2)=6\) and \(f^{\prime}(2)=5\) into \(y - f(a)=f^{\prime}(a)(x - a)\). We get \(y-6 = 5(x - 2)\).

Step4: Simplify the equation

Expand the right - hand side: \(y-6=5x-10\). Then add 6 to both sides: \(y = 5x-4\).

Answer:

\(y = 5x - 4\)