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What is a Constant Function?

What is a constant function?

Summary

  1. A constant function is a special type of linear function that follows the form f(x) = b
  2. 'b' is the y-intercept of the line and is just a constant
  3. A constant function is a linear function whose slope is 0
  4. No matter what value of 'x' you choose, the value of the function will always be the same
  5. The graph of a constant function is a horizontal line through the y-intercept, 'b'
  6. In example A, since b is 4, the graph is a horizontal line through y = 4
  7. In example B, since b is -2, the graph is a horizontal line through y = -2

Notes

    1. A linear function is a function whose graph is a line
    1. When you have a function with one variable like a linear function, f(x) and 'y' are the same thing
    2. The graph of a linear function is just a line
    3. The y-intercept is where the line crosses the y-axis
    1. 'm' stands for the slope of the line
    2. So let's say that 'm', our slope, is equal to 0
    3. Then we can plug 0 in for 'm' into our formula
    4. 0 times anything is 0, so the 'x' term cancels out
    5. Since 0 plus anything is just that number, 0+b leaves us with 'b'
    6. Remember, 'b' is just a constant: the y-intercept of the line
    1. Here 'b' must be a real number
    2. 'b' is also the y-intercept of the line
    1. Does this type of equation look familiar?
    2. Remember, a line with a slope of 0 is a horizontal line
    3. So the graphs of constant functions are just horizontal lines!
    1. In the first example f(x) is equal to 5, which is a constant
    2. So this is a constant function
    3. Constant functions can be equal to negative constants too, like the second example where b = -2
    1. Our function f(x) = 4 doesn't have any x's in it!
    2. So no matter what value we choose for 'x', we'll still get back 4
    1. Since there is no 'x' value in the function, it doesn't change when we choose different values for 'x'
    2. The value of the function will always remain constant
    3. That's why it's called a constant function!
    1. Remember, when we defined the function before we had f(x) = b, where 'b' was the y-intercept
    2. Our y-value is always going to be equal to that constant 'b'
    3. So the graph will be a horizontal line going through all the points where y = b
    1. This is just like a linear function with no 'x' term
    2. Remember, the 'x' term canceled out, since the slope was 0
    3. 'b' is equal to the y-intercept
    4. Since the y-value of the function is the same no matter what you choose for 'x', the graph is a horizontal line