Math Problem Statement

Copy and complete the table below and use it to plot the graph of y = x^2 + 3x - 2. Use your graph to estimate solutions to the equation x^2 + 3x - 2 = 0.

Solution

The problem asks you to complete the table for the quadratic function y=x2+3x−2y = x^2 + 3x - 2, and then use that information to plot the graph and estimate the solutions to x2+3x−2=0x^2 + 3x - 2 = 0.

Step 1: Complete the table by substituting values of xx into the function y=x2+3x−2y = x^2 + 3x - 2.

Let's calculate yy for each given xx:

y=x2+3x−2y = x^2 + 3x - 2

  • For x=−5x = -5: y=(−5)2+3(−5)−2=25−15−2=8y = (-5)^2 + 3(-5) - 2 = 25 - 15 - 2 = 8
  • For x=−4x = -4: y=(−4)2+3(−4)−2=16−12−2=2y = (-4)^2 + 3(-4) - 2 = 16 - 12 - 2 = 2
  • For x=−3x = -3: y=(−3)2+3(−3)−2=9−9−2=−2y = (-3)^2 + 3(-3) - 2 = 9 - 9 - 2 = -2
  • For x=−2x = -2: y=(−2)2+3(−2)−2=4−6−2=−4y = (-2)^2 + 3(-2) - 2 = 4 - 6 - 2 = -4
  • For x=−1x = -1: y=(−1)2+3(−1)−2=1−3−2=−4y = (-1)^2 + 3(-1) - 2 = 1 - 3 - 2 = -4
  • For x=0x = 0: y=(0)2+3(0)−2=−2y = (0)^2 + 3(0) - 2 = -2
  • For x=1x = 1: y=(1)2+3(1)−2=1+3−2=2y = (1)^2 + 3(1) - 2 = 1 + 3 - 2 = 2
  • For x=2x = 2: y=(2)2+3(2)−2=4+6−2=8y = (2)^2 + 3(2) - 2 = 4 + 6 - 2 = 8

Now, the completed table looks like this:

x & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 \\ \hline y & 8 & 2 & -2 & -4 & -4 & -2 & 2 & 8 \\ \end{array}$$ ### Step 2: Plot the graph of $$y = x^2 + 3x - 2$$ - Use the points $$(-5, 8)$$, $$(-4, 2)$$, $$(-3, -2)$$, $$(-2, -4)$$, $$(-1, -4)$$, $$(0, -2)$$, $$(1, 2)$$, and $$(2, 8)$$ to plot the graph. ### Step 3: Estimate the solutions to the equation $$x^2 + 3x - 2 = 0$$. The solutions to $$x^2 + 3x - 2 = 0$$ are the $$x$$-values where the graph crosses the $$x$$-axis (i.e., where $$y = 0$$). Based on the table and graph: - The graph crosses the $$x$$-axis approximately at $$x = -4$$ and $$x = 0.5$$. Thus, the estimated solutions are $$x \approx -4$$ and $$x \approx 0.5$$. Would you like further details or clarifications? Here are some related questions to consider: 1. What does the vertex of the parabola represent in this case? 2. How can you verify the exact solutions to the equation algebraically? 3. What is the axis of symmetry of the parabola? 4. How does the coefficient of $$x^2$$ affect the shape of the graph? 5. How can you factor $$x^2 + 3x - 2$$ to find the exact roots? **Tip**: When graphing quadratics, always look for the vertex and axis of symmetry to guide the shape of your graph.

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Graphing Parabolas

Formulas

y = x^2 + 3x - 2
Quadratic equation formula

Theorems

Graphing parabolas
Finding roots of quadratic equations

Suitable Grade Level

Grade 9

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