Math Problem Statement
Graph π(π₯) = π₯ 2 + 8π₯ + 12 using the vertex and finding the x-intercepts.
Solution
To graph the quadratic function , we can follow these steps:
1. Find the vertex
The vertex form of a quadratic equation is given by , where is the vertex of the parabola. We can find the vertex of a quadratic equation in standard form by using the formula:
For , the coefficients are:
Now, calculate the x-coordinate of the vertex:
To find the y-coordinate of the vertex, substitute into the original equation:
So, the vertex is at .
2. Find the x-intercepts
To find the x-intercepts, set and solve for :
Now, solve the quadratic equation using factoring:
Set each factor equal to zero:
Solving these gives:
So, the x-intercepts are at and .
3. Graph the function
Now that we know the following key points, we can sketch the graph:
- The vertex is at .
- The x-intercepts are at and .
Since the parabola opens upwards (because , which is positive), we can plot the vertex and x-intercepts, and then sketch the curve that passes through these points.
Summary of Key Points:
- Vertex:
- x-intercepts: and
- The parabola opens upwards.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Graphing Parabolas
Formulas
Vertex formula: x = -b/(2a)
Factoring of quadratic expressions
Theorems
Vertex of a parabola
X-intercepts of a quadratic function
Suitable Grade Level
Grades 8-10