Analyzing the table given in the exercise, you can identify that each input value (x-value) has one and only one output value (y-value). Therefore, the relation is a function.
Notice that "x" is the Independent Variable and "y" is the Dependent Variable.
• Observe that when:
[tex]x=0[/tex]
The value of "y" is:
[tex]y=9[/tex]
Therefore, the Start value is:
[tex]y=9[/tex]
• To find the Rate of change you can follow these steps:
- You can identify these ordered pairs:
[tex](0,9),(2,8),(4,7),(6,6)[/tex]
If you plot them on a Coordinate Plane, you get:
You can identify that it is a line. Therefore, the Rate to change can be found with this formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
In this case, you can choose these points:
[tex](0,9);(6,6)[/tex]
And set up that:
[tex]\begin{gathered} y_2=9 \\ y_1=6 \\ x_2=0 \\ x_1=6 \end{gathered}[/tex]
Then, substituting and evaluating, you get:
[tex]m=\frac{9-6}{0-6}=\frac{3}{-6}=-\frac{1}{2}[/tex]
• To find the equation for the relation, you need to remember that the
equation of a line can be written in Slope-Intercept Form:
[tex]y=mx+b[/tex]
Where "m" is the slope (the rate of change) and "b" is the y-intercept (the start value).
Since you already know both values, you can set up that the equation is:
[tex]y=-\frac{1}{2}x+9[/tex]
• To find the value of "y" when:
[tex]x=100[/tex]
You need to substitute that value into the equation and then evaluate:
[tex]\begin{gathered} y=(-\frac{1}{2})(100)+9 \\ \\ y=-50+9 \\ y=-41 \end{gathered}[/tex]
Hence, the answers are:
• Start value:
[tex]y=9[/tex]
• Rate of change:
[tex]m=-\frac{1}{2}[/tex]
• Relation:
[tex]y=-\frac{1}{2}x+9[/tex]
• Table:
