To determine which graph represents the function [tex]\( f(x) = x^2 + 3x + 2 \)[/tex], we need to analyze the key characteristics of the quadratic function. Let's go through these characteristics step-by-step.
1. Vertex:
The vertex of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] can be found using the formula [tex]\( x = \frac{-b}{2a} \)[/tex]. For [tex]\( f(x) = x^2 + 3x + 2 \)[/tex]:
[tex]\[
a = 1, \quad b = 3, \quad c = 2
\][/tex]
[tex]\[
x = \frac{-3}{2 \cdot 1} = -1.5
\][/tex]
Substituting [tex]\( x = -1.5 \)[/tex] back into the function to find the [tex]\( y \)[/tex]-coordinate of the vertex:
[tex]\[
y = (-1.5)^2 + 3(-1.5) + 2 = 2.25 - 4.5 + 2 = -0.25
\][/tex]
Therefore, the vertex is [tex]\( (-1.5, -0.25) \)[/tex].
2. Axis of Symmetry:
The axis of symmetry of the parabola is the vertical line that passes through the vertex, given by [tex]\( x = -1.5 \)[/tex].
3. Discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex]. For [tex]\( f(x) = x^2 + 3x + 2 \)[/tex]:
[tex]\[
\Delta = 3^2 - 4 \cdot 1 \cdot 2 = 9 - 8 = 1
\][/tex]
Since the discriminant is positive ([tex]\( \Delta > 0 \)[/tex]), there are two distinct real roots.
4. Roots:
The roots (x-intercepts) of the quadratic function can be found using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex]. For [tex]\( f(x) = x^2 + 3x + 2 \)[/tex]:
[tex]\[
x_1 = \frac{-3 + \sqrt{1}}{2 \cdot 1} = \frac{-3 + 1}{2} = -1
\][/tex]
[tex]\[
x_2 = \frac{-3 - \sqrt{1}}{2 \cdot 1} = \frac{-3 - 1}{2} = -2
\][/tex]
Therefore, the roots are [tex]\( x_1 = -1 \)[/tex] and [tex]\( x_2 = -2 \)[/tex].
Based on these calculations, we can determine that the graph of the function [tex]\( f(x) = x^2 + 3x + 2 \)[/tex] should exhibit the following key features:
- Vertex at [tex]\((-1.5, -0.25)\)[/tex].
- Axis of symmetry at [tex]\( x = -1.5 \)[/tex].
- Two distinct real roots at [tex]\( x = -1 \)[/tex] and [tex]\( x = -2 \)[/tex].
Among the given options, the correct graph that matches all these characteristics is graph 4.
Thus, the function [tex]\( f(x) = x^2 + 3x + 2 \)[/tex] is represented by:
D. graph 4.
