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 y>2x23x+1y > 2x^2 - 3x + 1, follow these steps:

1. Solve the inequality algebraically:

The given inequality is: y>2x23x+1y > 2x^2 - 3x + 1

  • This is a quadratic inequality, where the right-hand side is a parabola.
  • The first step is to treat the expression as an equation y=2x23x+1y = 2x^2 - 3x + 1 and find the critical points (vertex and x-intercepts).

2. Find the vertex:

For a quadratic equation of the form y=ax2+bx+cy = ax^2 + bx + c, the x-coordinate of the vertex is found using: xvertex=b2ax_{\text{vertex}} = \frac{-b}{2a} Here, a=2a = 2, b=3b = -3, and c=1c = 1.

xvertex=(3)2(2)=34x_{\text{vertex}} = \frac{-(-3)}{2(2)} = \frac{3}{4} Now, substitute xvertex=34x_{\text{vertex}} = \frac{3}{4} into the equation to find the y-coordinate of the vertex: yvertex=2(34)23(34)+1y_{\text{vertex}} = 2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) + 1 Calculating the result: yvertex=2(916)3(34)+1=181694+1=9894+1y_{\text{vertex}} = 2\left(\frac{9}{16}\right) - 3\left(\frac{3}{4}\right) + 1 = \frac{18}{16} - \frac{9}{4} + 1 = \frac{9}{8} - \frac{9}{4} + 1 yvertex=98188+88=18y_{\text{vertex}} = \frac{9}{8} - \frac{18}{8} + \frac{8}{8} = \frac{-1}{8} Thus, the vertex is at (34,18)\left( \frac{3}{4}, -\frac{1}{8} \right).

3. Plot the parabola:

  • The parabola opens upwards because a=2a = 2 (positive coefficient).
  • The vertex (34,18)\left( \frac{3}{4}, -\frac{1}{8} \right) is the minimum point.

4. Find the x-intercepts (if needed for graphing):

The x-intercepts occur when y=0y = 0, so solve: 0=2x23x+10 = 2x^2 - 3x + 1 Using the quadratic formula: x=(3)±(3)24(2)(1)2(2)=3±984=3±14x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(1)}}{2(2)} = \frac{3 \pm \sqrt{9 - 8}}{4} = \frac{3 \pm 1}{4} So the x-intercepts are: x1=3+14=1,x2=314=12x_1 = \frac{3 + 1}{4} = 1, \quad x_2 = \frac{3 - 1}{4} = \frac{1}{2} Thus, the x-intercepts are x=1x = 1 and x=12x = \frac{1}{2}.

5. Graph the inequality:

  • Plot the parabola using the vertex (34,18)\left( \frac{3}{4}, -\frac{1}{8} \right) and the x-intercepts x=1x = 1 and x=12x = \frac{1}{2}.
  • Since the inequality is y>2x23x+1y > 2x^2 - 3x + 1, you need to shade the region above the parabola.
  • The boundary parabola y=2x23x+1y = 2x^2 - 3x + 1 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 y>2x23x+1y > 2x^2 - 3x + 1.
  • 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:

  1. How does changing the coefficient of x2x^2 affect the direction and width of a parabola?
  2. What is the significance of the vertex in a quadratic inequality?
  3. How would the graph change if the inequality was y2x23x+1y \geq 2x^2 - 3x + 1?
  4. What are the differences between graphing a strict inequality and a non-strict inequality?
  5. 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