Math Problem Statement

  1. The figure shows the graph of the quadratic function y = x2 + bx + c, y where b and c are constants. (Given that have two pt (0,5), (6,5)), the graph opens upward) (a) Find the values of b and c. (b) Find the minimum value of y.

Solution

To solve the given problem, we need to find the constants bb and cc of the quadratic function y=x2+bx+cy = x^2 + bx + c, given the points (0,5)(0,5) and (6,5)(6,5).

Part (a): Find the values of bb and cc

  1. Using the point (0,5)(0,5):

    Since (0,5)(0,5) lies on the graph, we can substitute x=0x = 0 and y=5y = 5 into the equation:

    5=02+b(0)+c5 = 0^2 + b(0) + c c=5c = 5

  2. Using the point (6,5)(6,5):

    Since (6,5)(6,5) also lies on the graph, we substitute x=6x = 6 and y=5y = 5:

    5=62+b(6)+c5 = 6^2 + b(6) + c 5=36+6b+55 = 36 + 6b + 5 5=41+6b5 = 41 + 6b 6b=5416b = 5 - 41 6b=366b = -36 b=6b = -6

Therefore, the values of the constants are: b=6b = -6 c=5c = 5

Part (b): Find the minimum value of yy

The minimum value of yy for a quadratic function y=ax2+bx+cy = ax^2 + bx + c (where a>0a > 0) occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:

x=b2ax = -\frac{b}{2a}

For our quadratic function y=x26x+5y = x^2 - 6x + 5, a=1a = 1 and b=6b = -6:

x=62(1)=62=3x = -\frac{-6}{2(1)} = \frac{6}{2} = 3

To find the minimum value of yy, we substitute x=3x = 3 back into the quadratic equation:

y=(3)26(3)+5y = (3)^2 - 6(3) + 5 y=918+5y = 9 - 18 + 5 y=4y = -4

Therefore, the minimum value of yy is: y=4y = -4

Do you want any further details or have any questions?

Here are some questions you might want to ask next:

  1. How do you find the vertex of a quadratic function?
  2. What is the significance of the vertex in determining the minimum or maximum value of a quadratic function?
  3. How can you determine if a quadratic function opens upwards or downwards?
  4. What are some methods to solve quadratic equations?
  5. How does the value of the discriminant affect the nature of the roots of a quadratic equation?

Tip: When working with quadratic functions, identifying key points such as the vertex and the intercepts can provide valuable insights into the graph and help solve related problems effectively.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex of a Parabola
Minimum Value of Quadratic Function

Formulas

Quadratic equation formula: y = ax^2 + bx + c
Vertex formula: x = -b / (2a)
Minimum value formula: y = f(-b / (2a))

Theorems

-

Suitable Grade Level

High School