Mathematics » Differential Calculus » Sketching Graphs

# Interpreting Graphs

## Example

### Question

Consider the graph of the derivative of $$g(x)$$. 1. For which values of $$x$$ is $$g(x)$$ decreasing?
2. Determine the $$x$$-coordinate(s) of the turning point(s) of $$g(x)$$.
3. Given that $$g(x) = ax^{3} + bx^{2} + cx$$, calculate $$a$$, $$b$$ and $$c$$.

### Examine the parabolic graph and interpret the given information

We know that $$g'(x)$$ describes the gradient of $$g(x)$$. To determine where the cubic function is decreasing, we must find the values of $$x$$ for which $$g'(x) < 0$$:

$\{x: – 2 < x < 1; x \in \mathbb{R} \} \text{ or we can write } x \in (-2;1)$ ### Determine the $$x$$-coordinate(s) of the turning point(s)

To determine the turning points of a cubic function, we let $$g'(x) = 0$$ and solve for the $$x$$-values. These $$x$$-values are the $$x$$-intercepts of the parabola and are indicated on the given graph:

$x = -2 \text{ or } x = 1$

### Determine the equation of $$g(x)$$

\begin{align*} g(x) &= ax^{3} + bx^{2} + cx \\ g'(x) &= 3ax^{2} + 2bx + c \end{align*}

From the graph, we see that the $$y$$-intercept of $$g'(x)$$ is $$-\text{6}$$.

\begin{align*} \therefore c &= – 6 \\ g'(x) &= 3ax^{2} + 2bx – 6 \\ \text{Substitute } x = -2: \enspace g'(-2) &= 3a(-2)^{2} + 2b(-2) – 6 \\ 0 &= 12a – 4b – 6 \ldots \ldots (1) \\ \text{Substitute } x = 1: \enspace g'(1) &= 3a(1)^{2} + 2b(1) – 6 \\ 0 &= 3a + 2b – 6 \ldots \ldots (2) \end{align*}\begin{align*} \text{Eqn. }(1) – 4 \text{ Eqn. }(2): \quad 0 &= 0 -12b + 18 \\ \therefore b &= \frac{3}{2} \\ \text{And } 0 &= 3a + 2 \left( \frac{3}{2} \right) – 6 \\ 0 &= 3a – 3 \\ \therefore a &= 1 \\ & \\ g(x) &= x^{3} + \frac{3}{2}x^{2} – 6x \end{align*} 