Question
Consider the function \( f(x) = e^x – 3x – 4 \).
(a) On the following axes, sketch the graph of \( f \) for \(-4 \leq x \leq 3\). [3]
(b) The function \( g \) is defined by \( g(x) = e^{2x} – 6x – 7 \). The graph of \( g \) is obtained from the graph of \( f \) by a horizontal stretch with scale factor \( k \), followed by a vertical translation of \( c \) units. Find the value of \( k \) and the value of \( c \). [2]
▶️Answer/Explanation
(a) The graph of \( f(x) = e^x – 3x – 4 \) should show: – A y-intercept at \( (0, -3) \). – A local minimum around \( x = 1 \). – Roots approximately at \( x = -1 \) and \( x = 2 \).
(b) The horizontal stretch factor \( k = \frac{1}{2} \) and the vertical translation \( c = -3 \).