Math Problem Statement
Use 1 whole sheet of white pad paper and a graphing paper to find the solution set and graph of y > 2x2 - 3x+ 1
Solution
To solve and graph the inequality , follow these steps:
1. Solve the inequality algebraically:
The given inequality is:
- This is a quadratic inequality, where the right-hand side is a parabola.
- The first step is to treat the expression as an equation and find the critical points (vertex and x-intercepts).
2. Find the vertex:
For a quadratic equation of the form , the x-coordinate of the vertex is found using: Here, , , and .
Now, substitute into the equation to find the y-coordinate of the vertex: Calculating the result: Thus, the vertex is at .
3. Plot the parabola:
- The parabola opens upwards because (positive coefficient).
- The vertex is the minimum point.
4. Find the x-intercepts (if needed for graphing):
The x-intercepts occur when , so solve: Using the quadratic formula: So the x-intercepts are: Thus, the x-intercepts are and .
5. Graph the inequality:
- Plot the parabola using the vertex and the x-intercepts and .
- Since the inequality is , you need to shade the region above the parabola.
- The boundary parabola is not included in the solution set, so draw a dashed curve for the parabola.
Graphing Tips:
- On the graph paper, plot the points for the vertex, the x-intercepts, and other points as needed to capture the shape of the parabola.
- Use a dashed line to represent the boundary of the inequality, as the points on the parabola do not satisfy the strict inequality .
- Shade the region above the parabola to represent the solution set.
Let me know if you would like further clarifications or if you want to see a step-by-step graph.
Here are 5 related questions to expand your understanding:
- How does changing the coefficient of affect the direction and width of a parabola?
- What is the significance of the vertex in a quadratic inequality?
- How would the graph change if the inequality was ?
- What are the differences between graphing a strict inequality and a non-strict inequality?
- How do you determine the region to shade when solving inequalities graphically?
Tip: Always pay attention to whether the inequality is strict or non-strict to know whether the boundary line or curve should be dashed or solid.
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Math Problem Analysis
Mathematical Concepts
Quadratic Inequalities
Graphing Parabolas
Algebra
Formulas
Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
Vertex Formula: x_vertex = -b / 2a
Theorems
Properties of Quadratic Functions
Graphing Quadratic Inequalities
Suitable Grade Level
Grades 9-11