I posted this question yesterday and I got a response that is correct but I still don't understand how he got it. What did he do with the 2?
how do i find the slope of the curve at the indicated point?
f(x)= absolute value of x, x=2
math - bobpursley, Wednesday, September 16, 2009 at 7:50pm
at x=2, abs(x)= x
f(x)=x
slope= 1
f(x) = │x│
or
y = │x│
resulting in two linear functions
y = x , y = -x , where y ≥ 0
the graph will be a V-shape with the vertex at the origin.
here are a couple of ordered pairs,
for x>0, (1,1),(3,3),.. (12,12) etc
for x< 0
(-1,1),(-3,3) etc
so for x>0 the slope is +1
for x<0 the slope is -1
To understand how the person got the slope of 1 at x=2 for the function f(x) = |x|, let's break down the process step-by-step:
1. Start with the function f(x) = |x|.
- The absolute value function, denoted by |x|, takes the input value x and returns its magnitude (or positive value), regardless of whether x is positive or negative.
2. Find the value of f(x) at the specific x-coordinate given, which in this case is x=2.
- So, we substitute x=2 into the original function: f(2) = |2| = 2.
- This means that at x=2, the value of f(x) is 2.
3. The slope of a curve at a specific point can be understood as the rate at which the function is changing at that point.
- To calculate the slope, we need to find how much f(x) changes as x changes, specifically around the given x-coordinate.
4. In this case, when x=2, the function is simply f(x) = x.
- This is because the absolute value function evaluates to its positive input when x≥0, and x=2 satisfies this condition.
- Therefore, f(x) = x.
5. Now, using the simplified function f(x) = x, we need to find the slope at x=2.
- The slope of a linear function (like f(x) = x) is constant and equal to the coefficient of x.
- In this case, the coefficient of x is 1.
- So, the slope of the function f(x) = x, at any given point, is 1.
6. Since the given point is x=2, the slope of the curve at x=2 is also 1, which is the answer.
In summary, the person used the fact that the absolute value function evaluates to its positive input when x≥0, simplified the function to f(x) = x, and found that the slope of a linear function is equal to its coefficient, which is 1 in this case.