To find the intercepts of the function $f(x) = (x + 3)(4x^2 - 1)$, let's break down the steps.

1. Finding the x-intercepts

The x-intercepts occur where $f(x) = 0$. So, we need to solve: $(x + 3)(4x^2 - 1) = 0$ Set each factor equal to zero:

• For $x + 3 = 0$: $x = -3$

• For $4x^2 - 1 = 0$: $4x^2 = 1$ $x^2 = \frac{1}{4}$ $x = \pm \frac{1}{2}$

So, the x-intercepts are at $x = -3$, $x = \frac{1}{2}$, and $x = -\frac{1}{2}$.

Therefore, the x-intercepts are the points: $(-3, 0), \left(\frac{1}{2}, 0\right), \text{ and } \left(-\frac{1}{2}, 0\right)$

2. Finding the y-intercept

The y-intercept occurs where $x = 0$. Substitute $x = 0$ into the function: $f(0) = (0 + 3)(4 \cdot 0^2 - 1) = 3 \cdot (-1) = -3$

So, the y-intercept is: $(0, -3)$

Summary of Intercepts:

• x-intercepts: $(-3, 0)$, $\left(\frac{1}{2}, 0\right)$, $\left(-\frac{1}{2}, 0\right)$
• y-intercept: $(0, -3)$

Would you like more details on each step, or have any questions?

Here are five related questions to expand your understanding:

1. How would the intercepts change if the function were $f(x) = (x - 2)(4x^2 - 1)$?
2. What is the significance of the x-intercepts in terms of the graph of the function?
3. How does factoring help in finding intercepts of polynomial functions?
4. What is the effect on the intercepts if we multiply the function by a constant?
5. How can you determine if the function has symmetry based on its equation?

Tip: When finding intercepts, always check each factor independently by setting them equal to zero, as each can contribute its own intercept!