The ordered pairs (0,-3), (1,2), (2,7), (3,12) represent a function. What is a rule that represents this function?
Note that y increases 5 when x increases 1. So, the rule will look like
y = 5x
But, when x=0, 5x=0 and we want y = -3.
So,
y = 5x-3
You have to look for some pattern
Did you notice that for every 1 unit increase in the x's there is a 5 unit increase in the y's
so you know it starts with
y = 5x + b, we still don't know the b
plug in one of the points , how about (1,2)
2 = 5(1) + b
b = -3
function:
y = 5x - 3 or
f(x) = 5x - 3
To determine the rule that represents this function, we need to identify the relationship between the input (x-values) and the output (y-values) in the ordered pairs.
Let's analyze the pattern:
For each input (x-value):
- When x = 0, y = -3.
- When x = 1, y = 2.
- When x = 2, y = 7.
- When x = 3, y = 12.
By observing the pattern, we notice that the y-values increase by 5 each time the x-value increases by 1. This tells us that the rule for this function is an arithmetic relationship, specifically, a linear function with a constant rate of change.
To represent this function using algebra, we can write it in the form of y = mx + b, where m represents the rate of change (slope) and b represents the y-intercept.
To calculate the slope (m):
- Select any two ordered pairs (x1, y1) and (x2, y2) from the given values. Let's choose (0, -3) and (1, 2).
- The slope formula is: m = (y2 - y1) / (x2 - x1).
- Substituting the values, we have: m = (2 - (-3)) / (1 - 0) = 5.
So, the slope (m) of this function is 5.
Next, we need to find the y-intercept (b) by substituting the values of any of the ordered pairs into the equation y = mx + b.
Using the ordered pair (0, -3), we have:
- y = mx + b
- -3 = 5(0) + b
- -3 = b
Therefore, the y-intercept (b) is -3.
Now, we have the slope (m = 5) and the y-intercept (b = -3). We can substitute these values into the equation y = mx + b to determine the rule that represents this function:
The rule for this function is y = 5x - 3.