Find the rule for the following function table.
When we’re working with a rule for a function table, we wanna use 𝑦 equals 𝑚𝑥 plus 𝑏. This equation is what helps us relate the inputs to the outputs.
In this formula, the 𝑚 represents the slope and the 𝑏 represents the 𝑦-intercept. Our 𝑥-value will be the input and our 𝑦-value will be the output. Let’s start by finding the slope.
Our slope is the changes in 𝑦 over the changes in 𝑥. How are our 𝑦-values, our outputs, changing compared to our inputs? From 10 to 14, our output has increased by four. From 14 to 18, our output increases again by four. And from 18 to 22 and then from 22 to 26, each time our output is increasing by four. At the same time, our input is increasing by one. Each time our input increases by one. We can say that our slope equals four over one; our 𝑦 is changing by four and our 𝑥 is changing by one.
Okay. Now we have our 𝑚, so we know that our formula will be 𝑦 equals four 𝑥 plus 𝑏, which we don’t know yet. To find 𝑏, we’ll pick one set of input and output values and plug them in for our 𝑥 and 𝑦. And then we’ll solve for 𝑏. Let’s use one and 10. So I’ve plugged in one for 𝑥 and 10 for 𝑦. Let’s figure out what 𝑏 is. Four times one equals four. We subtract four from both sides to isolate 𝑏. 10 minus four equals six, which lets us know that 𝑏, our 𝑦-intercept, equals six.
Now that we have our slope and our 𝑦-intercept, we can form our equation 𝑦 equals four 𝑥 plus six. Or simply, the rule of this function table equals four 𝑥 plus six.