Math Problem Statement

complete the graph of y, equals, left parenthesis, x, minus, 7, right parenthesis, squared, plus, 2y=(x−7) 2 +2.

Solution

To complete the graph of the equation:

2y=(x7)2+22y = (x - 7)^2 + 2

Steps to solve:

  1. Solve for yy: To express the equation in terms of yy, divide both sides by 2:

    y=(x7)2+22y = \frac{(x - 7)^2 + 2}{2}

    Simplifying:

    y=(x7)22+1y = \frac{(x - 7)^2}{2} + 1

  2. Understand the graph's features:

    • This is a parabola equation in the form of y=a(xh)2+ky = a(x - h)^2 + k, where the vertex is at (h,k)(h, k).
    • The vertex of this equation is at (7,1)(7, 1) because the equation is in the form (x7)2(x - 7)^2 and the constant outside is +1+1.
  3. Identify the direction of the parabola:

    • Since the coefficient of (x7)2(x - 7)^2 is positive (12\frac{1}{2}), the parabola opens upwards.
  4. Plot the vertex:

    • The vertex is the point (7,1)(7, 1).
  5. Plot a few other points:

    • When x=6x = 6, we have:

      y=(67)22+1=12+1=1.5y = \frac{(6 - 7)^2}{2} + 1 = \frac{1}{2} + 1 = 1.5

    • When x=8x = 8, we have:

      y=(87)22+1=12+1=1.5y = \frac{(8 - 7)^2}{2} + 1 = \frac{1}{2} + 1 = 1.5

    So the points (6,1.5)(6, 1.5) and (8,1.5)(8, 1.5) also lie on the parabola.

  6. Sketch the graph:

    • The graph is symmetric about the vertical line x=7x = 7 (the axis of symmetry).
    • The parabola opens upwards, and you can draw it passing through the vertex and the points (6,1.5)(6, 1.5) and (8,1.5)(8, 1.5).

Would you like a visual graph of this parabola or more details on the features?

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Parabolas

Formulas

y = (x - h)^2 + k
y = (x - 7)^2 / 2 + 1

Theorems

Vertex form of a quadratic equation
Properties of parabolas

Suitable Grade Level

Grades 8-10