Sure, let's analyze and sketch the graph of the function \( f(x) = \frac{1}{2} x^2 + 2x - 6 \). Here are the steps to create the graph:
### Step 1: Identify the Type of Function
The given function is a quadratic function in the form \( ax^2 + bx + c \). In this case:
- \( a = \frac{1}{2} \)
- \( b = 2 \)
- \( c = -6 \)
Since \( a > 0 \), the parabola opens upwards.
### Step 2: Find the Vertex
The vertex of a parabola \( ax^2 + bx + c \) is given by the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substitute the values of \( a \) and \( b \):
[tex]\[ x = -\frac{2}{2 \times \frac{1}{2}} = -\frac{2}{1} = -2 \][/tex]
Now, substitute \( x = -2 \) back into the function to find the y-coordinate of the vertex:
[tex]\[ f(-2) = \frac{1}{2}(-2)^2 + 2(-2) - 6 \][/tex]
[tex]\[ f(-2) = \frac{1}{2}(4) + (-4) - 6 \][/tex]
[tex]\[ f(-2) = 2 - 4 - 6 = -8 \][/tex]
So, the vertex is \((-2, -8)\).
### Step 3: Find the y-intercept
The y-intercept of the function is the value of \( f(x) \) at \( x = 0 \):
[tex]\[ f(0) = \frac{1}{2}(0)^2 + 2(0) - 6 = -6 \][/tex]
So, the y-intercept is \((0, -6)\).
### Step 4: Find the x-intercepts
The x-intercepts (or roots) are found by solving \( f(x) = 0 \):
[tex]\[ \frac{1}{2} x^2 + 2x - 6 = 0 \][/tex]
Multiply through by 2 to simplify:
[tex]\[ x^2 + 4x - 12 = 0 \][/tex]
Now, factor the quadratic equation:
[tex]\[ (x + 6)(x - 2) = 0 \][/tex]
So, the roots are:
[tex]\[ x = -6 \quad \text{and} \quad x = 2 \][/tex]
Thus, the x-intercepts are \((-6, 0)\) and \((2, 0)\).
### Step 5: Sketch the Graph
Using the information we have:
- The vertex is at \((-2, -8)\)
- The y-intercept is at \((0, -6)\)
- The x-intercepts are at \((-6, 0)\) and \((2, 0)\)
We can sketch the graph of the function.
Here's a step-by-step outline of how it will look:
1. Plot the vertex at \((-2, -8)\).
2. Plot the y-intercept at \((0, -6)\).
3. Plot the x-intercepts at \((-6, 0)\) and \((2, 0)\).
4. Draw a smooth curve through these points, forming a parabola that opens upwards because \(a = \frac{1}{2} > 0\).
### Conclusion
The graph of the function [tex]\( f(x) = \frac{1}{2} x^2 + 2x - 6 \)[/tex] is a parabola that opens upwards with its vertex at [tex]\((-2, -8)\)[/tex], y-intercept at [tex]\((0, -6)\)[/tex], and x-intercepts at [tex]\((-6, 0)\)[/tex] and [tex]\((2, 0)\)[/tex].
