Question

which equation is a function that is discrete, and has an absolute maximum?
y = x² - 8, where x in an integer
y = -x² + 8, where x in an integer
y = x² + 8, where x in an integer
y = -x + 8, where x in an integer

use the graphing tool to graph the function.

Explanation:

Step1: Understand Discrete Function

A discrete function has a domain (x - values) that is a set of distinct points (here, integers). So all options have discrete domains since \( x \) is an integer.

Step2: Analyze Absolute Maximum

An absolute maximum is the highest \( y \) - value the function attains.

• For \( y = x^{2}-8 \): The function \( y = x^{2}-8 \) is a parabola opening upwards (\( a = 1>0 \)). As \( |x|\) increases, \( y=x^{2}-8 \) increases without bound. So it has no absolute maximum, only an absolute minimum.
• For \( y=-x^{2}+8 \): The function \( y = - x^{2}+8 \) is a parabola opening downwards (\( a=- 1<0 \)). The vertex of the parabola \( y = ax^{2}+bx + c \) is at \( x =-\frac{b}{2a}\). Here \( b = 0\), \( a=-1 \), so \( x = 0\). When \( x = 0\), \( y=8\). As \( |x|\) increases (since \( x \) is an integer), \( y=-x^{2}+8\) decreases (because \( x^{2}\) is positive and we have a negative sign in front). So this function has an absolute maximum at \( x = 0\) ( \( y = 8\) ).
• For \( y=x^{2}+8 \): The function \( y=x^{2}+8 \) is a parabola opening upwards (\( a = 1>0 \)). As \( |x|\) increases, \( y=x^{2}+8\) increases without bound. So it has no absolute maximum, only an absolute minimum.
• For \( y=-x + 8\): This is a linear function with a slope of \( - 1\). As \( x\to-\infty\), \( y=-x + 8\to+\infty\) (since \( -x\) becomes very large positive when \( x\) is very large negative). So this function has no absolute maximum (it increases without bound as \( x\) decreases).

Answer:

\( y=-x^{2}+8 \), where \( x \) is an integer