write a function rule for the table.
y=
x -10, -7, -2, 6, 10
y -25, -19, -9, 7, 15
Can someone please show me step by step how to work this problem out step by step, I am so confused.
the slope is ∆y/∆x = (-19+25)/(-7+10) = 2
So start with y=2x
But, 2(-10) = -20, not -25
So adjust by subtracting 5, and you have
y = 2x-5
To find the function rule for the table, we need to determine the relationship between the input values (x) and the output values (y). Let's break it down step by step:
Step 1: Look for a pattern in the x and y values.
By comparing the x and y values, we can see that as x increases, y also increases. There seems to be a consistent increase in the y values for each corresponding x value.
Step 2: Calculate the difference between consecutive y values.
To determine the pattern more precisely, we can calculate the difference between consecutive y values:
- The difference between -25 and -19 is +6.
- The difference between -19 and -9 is +10.
- The difference between -9 and 7 is +16.
- The difference between 7 and 15 is +8.
Step 3: Calculate the difference between consecutive x values.
Next, let's calculate the difference between consecutive x values:
- The difference between -10 and -7 is +3.
- The difference between -7 and -2 is +5.
- The difference between -2 and 6 is +8.
- The difference between 6 and 10 is +4.
Step 4: Compare the differences.
Now, compare the differences between the x and y values. We can observe that the differences between the y values (+6, +10, +16, +8) are all multiples of 2, whereas the differences between the x values (+3, +5, +8, +4) are not multiples of 2.
Step 5: Determine the function rule.
Based on the analysis of the differences, we can conclude that the relationship between x and y is nonlinear. It is not a simple linear relationship such as y = mx + b, where m is the slope and b is the y-intercept.
The function rule for this table is best represented by a quadratic equation, which has the form y = ax^2 + bx + c.
To find the coefficients a, b, and c, we can use the values from the table:
We can start by plugging in the x and y values into the quadratic equation:
For x = -10, y = -25: -25 = a*(-10)^2 + b*(-10) + c
For x = -7, y = -19: -19 = a*(-7)^2 + b*(-7) + c
For x = -2, y = -9: -9 = a*(-2)^2 + b*(-2) + c
For x = 6, y = 7: 7 = a*6^2 + b*6 + c
For x = 10, y = 15: 15 = a*10^2 + b*10 + c
By solving this system of equations, we can find the values of a, b, and c, which will give us the specific function rule for the table.