Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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].
### 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].
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
