Alright, let's break down and graph the function [tex]\( y = (x-2)(x+3) \)[/tex] step by step to determine its key points and overall shape. Here are the important details:
### Step-by-step Solution:
1. Identify the Function Type:
The given function [tex]\( y = (x - 2)(x + 3) \)[/tex] is a quadratic function. In standard quadratic form, it can be written as [tex]\( y = ax^2 + bx + c \)[/tex].
2. Expand the Function:
Expanding [tex]\( y = (x - 2)(x + 3) \)[/tex]:
[tex]\[
y = x^2 + 3x - 2x - 6
\][/tex]
Combining like terms:
[tex]\[
y = x^2 + x - 6
\][/tex]
So, the quadratic equation is [tex]\( y = x^2 + x - 6 \)[/tex].
3. Identify the Coefficients:
From the equation [tex]\( y = x^2 + x - 6 \)[/tex], the coefficients are:
[tex]\[
a = 1, \quad b = 1, \quad c = -6
\][/tex]
4. Find the Roots (x-intercepts):
To find the roots of the function, set [tex]\( y \)[/tex] to 0 and solve for [tex]\( x \)[/tex]:
[tex]\[
0 = (x - 2)(x + 3)
\][/tex]
This gives us the roots:
[tex]\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\][/tex]
[tex]\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\][/tex]
So, the roots of the function are [tex]\( x = 2 \)[/tex] and [tex]\( x = -3 \)[/tex].
5. Find the Vertex:
The vertex of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[
x_{\text{vertex}} = -\frac{b}{2a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x_{\text{vertex}} = -\frac{1}{2 \cdot 1} = -\frac{1}{2}
\][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = -\frac{1}{2} \)[/tex] back into the function:
[tex]\[
y = \left(-\frac{1}{2} - 2\right)\left(-\frac{1}{2} + 3\right)
\][/tex]
[tex]\[
y = \left(-\frac{5}{2}\right)\left(\frac{5}{2}\right)
\][/tex]
[tex]\[
y = -\frac{25}{4} = -6.25
\][/tex]
So, the vertex is at [tex]\( \left( -0.5, -6.25 \right) \)[/tex].
6. Important Points to Consider:
- Roots (x-intercepts): [tex]\( x = 2 \)[/tex] and [tex]\( x = -3 \)[/tex]
- Vertex: [tex]\( \left( -0.5, -6.25 \right) \)[/tex]
### Graph Description:
- The graph is a parabola opening upwards (since [tex]\( a = 1 > 0 \)[/tex]).
- It crosses the x-axis at [tex]\( x = 2 \)[/tex] and [tex]\( x = -3 \)[/tex].
- The vertex of the parabola is at [tex]\( \left( -0.5, -6.25 \right) \)[/tex].
Make sure that the chosen answer accurately describes a parabola with these characteristics, specifically the roots and the vertex. Depending on the given answer choices, select the one that matches this graph.
