To determine the best minimum x- and minimum y-values for the viewing window when graphing the equations \( y = x^2 - 14x + 24 \) and \( y = 20 - |x + 3| \), we need to analyze both functions.
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Finding the intercepts of \( y = x^2 - 14x + 24 \):
- For x-intercepts, set \( y = 0 \): \[ x^2 - 14x + 24 = 0 \] Factoring gives \( (x - 12)(x - 2) = 0 \), so the x-intercepts are \( x = 2 \) and \( x = 12 \).
- For the y-intercept, set \( x = 0 \): \[ y = 0^2 - 14(0) + 24 = 24 \] Thus, the y-intercept is \( y = 24 \).
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Finding the minimum value of \( y = x^2 - 14x + 24 \):
- The vertex of the parabola \( y = ax^2 + bx + c \) is located at \( x = -\frac{b}{2a} \): \[ x = -\frac{-14}{2 \cdot 1} = 7 \]
- Substituting \( x = 7 \) back into the equation: \[ y = 7^2 - 14(7) + 24 = 49 - 98 + 24 = -25 \] So, the minimum value is \( y = -25 \) at \( x = 7 \).
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Finding the intercepts of \( y = 20 - |x + 3| \):
- For the x-intercepts, set \( y = 0 \): \[ 20 - |x + 3| = 0 \implies |x + 3| = 20 \] This gives two equations: \[ x + 3 = 20 \quad \Rightarrow \quad x = 17 \] \[ x + 3 = -20 \quad \Rightarrow \quad x = -23 \] Thus, the x-intercepts are \( x = 17 \) and \( x = -23 \).
- For the y-intercept, set \( x = 0 \): \[ y = 20 - |0 + 3| = 20 - 3 = 17 \] So, the y-intercept is \( y = 17 \).
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Finding the maximum value of \( y = 20 - |x + 3| \):
- The maximum value occurs at \( x = -3 \): \[ y = 20 - | -3 + 3 | = 20 - 0 = 20 \]
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Summary of values:
- x-intercepts of the quadratic: \( x = 2, 12 \)
- Minimum of the quadratic: \( y = -25 \) at \( x = 7 \)
- x-intercepts of \( y = 20 - |x + 3| \): \( x = -23, 17 \)
- Maximum of \( y = 20 - |x + 3| \): \( y = 20 \) at \( x = -3 \)
- y-intercepts: \( y = 24 \) for the quadratic, \( y = 17 \) for \( y = 20 - |x + 3| \)
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Determining the minimum viewing window:
- For the x-values: The smallest x-intercept is \( -23 \) and the largest x-intercept is \( 17 \). Therefore, the minimum x-value can be \( -25 \) to ensure visibility.
- For the y-values: The minimum y-value is \( -25 \) from the quadratic, and the maximum y-value is \( 24 \). Therefore, the minimum y-value can also be \( -25 \).
Based on this analysis, the best minimum x-value is \( -25 \) which provides the required view to see the relevant points of interest in both graphs. Thus, the recommended values for the viewing window are:
- Minimum x-value: –25
- Minimum y-value: –25
Therefore, the answer is –25 for the minimum x-value.