Math Problem Statement

Sketch the graph of Y= -x^(2) +6x - 8

Solution

To sketch the graph of the function y=x2+6x8y = -x^2 + 6x - 8, we'll follow a similar step-by-step approach to analyze its key features, including the vertex, axis of symmetry, intercepts, and overall shape.

Step-by-Step Analysis

  1. Identify the Type of Function:
    The function y=x2+6x8y = -x^2 + 6x - 8 is a quadratic function. Its general form is y=ax2+bx+cy = ax^2 + bx + c, where a=1a = -1, b=6b = 6, and c=8c = -8.

  2. Determine the Shape:
    Since a=1<0a = -1 < 0, the parabola opens downwards.

  3. Find the Vertex:
    The vertex form of a parabola is given by x=b2ax = -\frac{b}{2a}. For this function: x=62×(1)=62=3.x = -\frac{6}{2 \times (-1)} = \frac{-6}{-2} = 3. To find the y-coordinate of the vertex, substitute x=3x = 3 into the function: y=(3)2+6(3)8=9+188=1.y = -(3)^2 + 6(3) - 8 = -9 + 18 - 8 = 1. So, the vertex is at (3,1)(3, 1).

  4. Axis of Symmetry:
    The axis of symmetry for a parabola y=ax2+bx+cy = ax^2 + bx + c is the vertical line passing through the vertex. In this case, it is x=3x = 3.

  5. Find the Y-intercept:
    The y-intercept occurs when x=0x = 0: y=(0)2+6(0)8=8.y = -(0)^2 + 6(0) - 8 = -8. So, the y-intercept is at (0,8)(0, -8).

  6. Find the X-intercepts:
    The x-intercepts occur when y=0y = 0. We solve the quadratic equation: x2+6x8=0.-x^2 + 6x - 8 = 0. Multiply the equation by -1 to simplify: x26x+8=0.x^2 - 6x + 8 = 0. Factor the quadratic: (x2)(x4)=0.(x - 2)(x - 4) = 0. So, x=2x = 2 and x=4x = 4. The x-intercepts are at (2,0)(2, 0) and (4,0)(4, 0).

  7. Sketch the Graph:

    • Plot the vertex at (3,1)(3, 1).
    • The parabola opens downwards.
    • The axis of symmetry is the vertical line x=3x = 3.
    • Plot the y-intercept at (0,8)(0, -8).
    • Plot the x-intercepts at (2,0)(2, 0) and (4,0)(4, 0).

Graph of the Function

Based on the analysis above, the graph is a downward-opening parabola with the vertex at (3,1)(3, 1), x-intercepts at (2,0)(2, 0) and (4,0)(4, 0), and a y-intercept at (0,8)(0, -8).

If you'd like a visual representation, here is a sketch of the graph:

  y
  |
  |           *
  |        *     *
  |     *           *
  |  *                 *
  |                         *      
--+--------------------------- x
  |                     *
  |                *
  |          *
  |     *
  |  *
  |

Would you like more details on any step, or have any specific questions?

Related Questions:

  1. How would the graph change if the equation were y=x26x+8y = x^2 - 6x + 8?
  2. What is the importance of finding the vertex in a quadratic function?
  3. How do you determine the direction in which a parabola opens?
  4. What is the role of the discriminant in finding the x-intercepts of a quadratic function?
  5. How can you convert a quadratic function from standard form to vertex form?

Tip: Knowing how to factor quadratic expressions can greatly simplify finding the x-intercepts of a quadratic function.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Vertex Form
Graphing

Formulas

Quadratic equation formula ax^2 + bx + c = 0
Vertex form formula x = -b/(2a)
Axis of symmetry formula x = -b/(2a)

Theorems

-

Suitable Grade Level

Grades 9-10