Write the equation that represents the linear function shown in the given table.
Since we know that this is a linear function — that is, it’s a straight line function — we know that 𝑦 equals 𝑚𝑥 plus 𝑏 is a general form of a linear function. And to find an equation, we need to know two things. We need the slope and the 𝑦-intercept. And the 𝑚-value, the slope, can be found as long as you have two points along the line. If you have two points, you can say 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one is the slope.
And finally, we know that the 𝑦-intercept is where 𝑥 equals zero. Bringing this information to the table, we see that when 𝑥 equals zero, 𝑦 equals five. The point zero, five falls on this line. And we can plug in the value five for 𝑏. Five is the 𝑦-intercept. We can list the other coordinates — negative two, three and negative one, four.
Remember that I said we just need two points to find the slope. We could use negative two, three; negative one, four. Or we could use negative one, four and zero, five or even negative two, three and zero, five. Any combination of these three points will work. Let’s say we take the first two. We’ll let the first point be 𝑥 one, 𝑦 one and the second point be 𝑥 two, 𝑦 two. Then the slope will be four minus three over negative one minus negative two. Four minus three is one. Negative one minus negative two is negative one plus two, which is one.
And so, we can say that the slope of this line is one. And 𝑦 will be equal to one times 𝑥 plus five, which we can simplify to be 𝑦 equals 𝑥 plus five. Now that we have this equation, let’s double-check it against the table. If we plug in negative two, negative two plus five does equal three. Negative one plus five does equal four. And zero plus five does equal five. Which confirms that we have found the correct linear function equation, 𝑦 equals 𝑥 plus five.