Math Problem Statement

Copy and complete the table of values for y = x^2 - 5x + 2. Plot the graph of y = x^2 - 5x + 2, and solve the equation x^2 - 5x + 2 = 2 by adding the line y = 2 to your diagram.

Solution

Let's solve this step by step:

1. Complete the table of values for y=x2−5x+2y = x^2 - 5x + 2:

We substitute each value of xx into the equation y=x2−5x+2y = x^2 - 5x + 2:

  • For x=−1x = -1:
    y=(−1)2−5(−1)+2=1+5+2=8y = (-1)^2 - 5(-1) + 2 = 1 + 5 + 2 = 8 (already given)

  • For x=0x = 0:
    y=02−5(0)+2=0−0+2=2y = 0^2 - 5(0) + 2 = 0 - 0 + 2 = 2 (already given)

  • For x=1x = 1:
    y=12−5(1)+2=1−5+2=−2y = 1^2 - 5(1) + 2 = 1 - 5 + 2 = -2

  • For x=2x = 2:
    y=22−5(2)+2=4−10+2=−4y = 2^2 - 5(2) + 2 = 4 - 10 + 2 = -4

  • For x=3x = 3:
    y=32−5(3)+2=9−15+2=−4y = 3^2 - 5(3) + 2 = 9 - 15 + 2 = -4

  • For x=4x = 4:
    y=42−5(4)+2=16−20+2=−2y = 4^2 - 5(4) + 2 = 16 - 20 + 2 = -2

  • For x=5x = 5:
    y=52−5(5)+2=25−25+2=2y = 5^2 - 5(5) + 2 = 25 - 25 + 2 = 2

  • For x=6x = 6:
    y=62−5(6)+2=36−30+2=8y = 6^2 - 5(6) + 2 = 36 - 30 + 2 = 8

Completed Table:

\hline x & y \\ \hline -1 & 8 \\ 0 & 2 \\ 1 & -2 \\ 2 & -4 \\ 3 & -4 \\ 4 & -2 \\ 5 & 2 \\ 6 & 8 \\ \hline \end{array}$$ --- #### 2. Plot the graph of $$y = x^2 - 5x + 2$$ using the table. Using the completed table, plot the points $$(-1, 8)$$, $$(0, 2)$$, $$(1, -2)$$, etc., on graph paper. Draw a smooth curve through these points, as the graph is a parabola. --- #### 3. Add the line $$y = 2$$ to the diagram: Draw a horizontal line through $$y = 2$$ on the graph. This line represents the equation $$y = 2$$. --- #### 4. Solve $$x^2 - 5x + 2 = 2$$ graphically: To find the solutions, determine where the graph of $$y = x^2 - 5x + 2$$ intersects the line $$y = 2$$. Set the equations equal: $$x^2 - 5x + 2 = 2$$ Simplify: $$x^2 - 5x = 0$$ Factorize: $$x(x - 5) = 0$$ Thus, the solutions are: $$x = 0 \quad \text{and} \quad x = 5$$ --- Would you like me to generate the graph, explain further, or assist with related questions? --- ### 5 Relative Questions: 1. What is the vertex of the parabola $$y = x^2 - 5x + 2$$? 2. How do we find the axis of symmetry for a quadratic function? 3. What are the other methods to solve $$x^2 - 5x + 2 = 2$$ (e.g., algebraically)? 4. Can you explain why the solutions to $$x^2 - 5x = 0$$ are $$x = 0$$ and $$x = 5$$? 5. How does the shape of the parabola change for different quadratic coefficients? ### Tip: When solving quadratic equations graphically, always look for the intersection points of the two curves or lines—these are the solutions.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Graphing Parabolas
Solving Quadratic Equations
Intersection of Functions

Formulas

y = x^2 - 5x + 2
Setting equations equal: x^2 - 5x + 2 = 2
Factoring: x(x - 5) = 0

Theorems

Zero Product Property

Suitable Grade Level

Grades 8-10

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