The curve C has equation y = f(x), where f'(x) = (x - 3)(3x + 5) Given that the point P (1, 20) lies on C, (a) find f(x), simplifying each term - Edexcel - A-Level Maths Pure - Question 10 - 2018 - Paper 1

To show this, we can expand the expression and find the value of A. Starting with: egin{align*} f(x) &= (x - 3)(x + A) \&= x^2 + Ax - 3x - 3A \&= x^2 + (A - 3)x - 3A. \end{align*}

We know: $f(x) = x^3 - 2x^2 - 15x + 36.$

To confirm equality, we need to match coefficients:

• The coefficient of $x^2$: 1 and -2.
• The coefficient of $x$: (A - 3) must equal -15.
• The constant: -3A must equal 36.

From A - 3 = -15, we get: $A = -15 + 3 = -12.$

From -3A = 36, we find: $A = -12.$

Thus, shown: $f(x) = (x - 3)(x - 12).$

To sketch the graph of C:

1. Finding Roots: To find where C meets the x-axis, set f(x) = 0: $f(x) = (x - 3)(x - 12) = 0.$ Thus, the points where C cuts the x-axis are at x = 3 and x = 12.

2. Finding Y-intercept: To find the y-intercept where C cuts the y-axis, we evaluate f(0): $f(0) = 36.$ Therefore, the point is (0, 36).

3. Sketch: The graph should be a cubic function that starts from the bottom left quadrant, rises through the y-axis at (0, 36), and crosses the x-axis at (3, 0) and (12, 0). Make sure to label these points clearly on the graph.