Let ƒ(x) = (x - 3) (x + 3)²


f. State the x- and y-intercepts of the graph of ƒ.

Verified step by step guidance

1. To find the x-intercepts, set ƒ(x) = 0 and solve for x. This means solving the equation (x - 3)(x + 3)² = 0.
2. The equation (x - 3)(x + 3)² = 0 can be solved by setting each factor equal to zero: x - 3 = 0 and (x + 3)² = 0.
3. Solving x - 3 = 0 gives x = 3. Solving (x + 3)² = 0 gives x = -3. Therefore, the x-intercepts are x = 3 and x = -3.
4. To find the y-intercept, evaluate ƒ(x) at x = 0. Substitute x = 0 into the function: ƒ(0) = (0 - 3)(0 + 3)².
5. Calculate ƒ(0) using the expression: ƒ(0) = (-3)(3)². This will give you the y-intercept, which is the point where the graph crosses the y-axis.

Key Concepts

X-Intercepts

X-intercepts are the points where the graph of a function crosses the x-axis. To find the x-intercepts, we set the function ƒ(x) equal to zero and solve for x. In this case, we solve (x - 3)(x + 3)² = 0, which gives us the values of x where the function equals zero.

Y-Intercepts

Y-intercepts are the points where the graph of a function crosses the y-axis. To find the y-intercept, we evaluate the function at x = 0. For the function ƒ(x) = (x - 3)(x + 3)², substituting x = 0 allows us to calculate the corresponding y-value, giving us the y-intercept.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. In the function ƒ(x) = (x - 3)(x + 3)², recognizing the factors helps in identifying the roots of the polynomial, which are essential for determining both x- and y-intercepts. Understanding how to factor is crucial for simplifying expressions and solving equations in calculus.