To determine the [tex]\( x \)[/tex]-intercepts of the quadratic function [tex]\( f(x) = 3(x-1)(x+3) \)[/tex], we need to find the points where the function [tex]\( f(x) \)[/tex] equals zero. These points occur when either of the factors [tex]\( (x-1) \)[/tex] or [tex]\( (x+3) \)[/tex] equal zero.
Let's solve for the [tex]\( x \)[/tex]-intercepts step-by-step:
1. Identify the factors: The quadratic function is given in factored form as [tex]\( f(x) = 3(x-1)(x+3) \)[/tex].
2. Set the function equal to zero: To find the [tex]\( x \)[/tex]-intercepts, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
3(x-1)(x+3) = 0
\][/tex]
3. Solve for each factor equal to zero:
- First factor: [tex]\( x-1 = 0 \)[/tex]
[tex]\[
x-1 = 0 \implies x = 1
\][/tex]
- Second factor: [tex]\( x+3 = 0 \)[/tex]
[tex]\[
x+3 = 0 \implies x = -3
\][/tex]
4. Determine the [tex]\( x \)[/tex]-intercepts: The solutions [tex]\( x = 1 \)[/tex] and [tex]\( x = -3 \)[/tex] are the values where the function crosses the [tex]\( x \)[/tex]-axis. Therefore, the corresponding [tex]\( x \)[/tex]-intercepts are [tex]\( (1, 0) \)[/tex] and [tex]\( (-3, 0) \)[/tex].
So, the correct [tex]\( x \)[/tex]-intercepts for the graph of the quadratic function [tex]\( f(x) = 3(x-1)(x+3) \)[/tex] are [tex]\( (1, 0) \)[/tex] and [tex]\( (-3, 0) \)[/tex].
Looking at the answer choices:
- [tex]\((3, 0)\)[/tex]
- [tex]\((-3, 0)\)[/tex] ✅
- [tex]\((-1, 0)\)[/tex]
- [tex]\((1, 0)\)[/tex] ✅
- [tex]\((3, 1)\)[/tex]
- [tex]\((-3, -1)\)[/tex]
The correct [tex]\( x \)[/tex]-intercepts are [tex]\((-3, 0)\)[/tex] and [tex]\((1, 0)\)[/tex]. Therefore, the correct answers are:
- [tex]\((-3, 0)\)[/tex]
- [tex]\((1, 0)\)[/tex]
