Final answer:
To ensure all intercepts, minimums, maximums, and points of intersection are visible on the graph of the equations y = x^2 - 14x + 24 and y = 20 - 5x + 31, the best minimum x-value for the viewing window is -5, and the best minimum y-value is -10.
Step-by-step explanation:
To ensure that all intercepts, minimums, maximums, and points of intersection are visible on the graph of the equations y = x^2 - 14x + 24 and y = 20 - 5x + 31, we need to determine the minimum x- and y-values for the viewing window. Here's how we can find these values:
- Find the x-intercepts of each equation by setting y = 0 and solving for x:
- y = x^2 - 14x + 24
- x^2 - 14x + 24 = 0
- Factor the equation: (x - 2)(x - 12) = 0
- x = 2 or x = 12
- The x-intercepts are x = 2 and x = 12
Find the y-intercepts of each equation by setting x = 0:
- y = x^2 - 14x + 24
- y = 0^2 - 14(0) + 24 = 24
- The y-intercept is y = 24
Find the values of y at the vertex of each equation:
- The vertex of the equation y = x^2 - 14x + 24 occurs at x = -b / (2a), where a and b are the coefficients of x^2 and x respectively.
- x = -(-14) / (2 * 1) = 7
- Substitute x = 7 into the equation to find y: y = (7)^2 - 14(7) + 24 = -1
- The vertex is (7, -1)
Determine the minimum x- and y-values for the viewing window:
- x-min: Choose a value to the left of the x-intercepts, such as -5.
- x-max: Choose a value to the right of the x-intercepts, such as 15.
- y-min: Choose a value below the y-intercept, such as -10.
- y-max: Choose a value above the vertex y-value, such as 5.
Therefore, the best minimum x-value for the viewing window is -5, and the best minimum y-value is -10.