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Intercepts and Asympototes
An intercept is where a function crosses a given axis.
Y-Intercept : This is where the equation crosses the y-axis. In order to cross the y-axis, the x-coordinate must be zero. (Insert random graphs) Think about it. When you look at the y-intercept in each graph, what is your x-value? The y-intercept is the y-value when the x-coordinate is zero.
How to solve for the y-intercept? As we said above, the y-intercept is the y-value when the x-coordinate is zero. It follows that if we want to find what the y-intercept of a function, we plug in zero for our x-value and calculate our y- value. The answer will be where the function crosses the y-axis. You will often write your answer as a coordinate pair: (0,y).
Sample Problem
1.) f(x)=x^3 +x^2 +
The y-intercept is the function value when x equals zero so,
Y-intercept=f(0)= (0)^3 +(0)^2 +2=2 so the coordinate pair is (0,2).
X-Intercept : This is where the equation crosses the x-axis. Similar to the y-intercept, in order to cross the x-axis, the y-coordinate must be zero. (Insert random graphs)
How to solve for the x-intercept? As we said above, the x-intercept is the coordinate pair when the y-coordinate is zero. It follows that if we want to find what the x-intercept of a function, we plug in zero for our y-value and solve for our x-value. The answer will be where the function crosses the x-axis. The coordinate pair will have the following form: (x,0).
Sample Problem
2.) f(x)= x^2 +3x+
The x-intercept is the x-value when y equals zero so,
f(x)=0=x^2 +3x+2=(x+1)(x+2)
For this equation to be true, x must equal -1 or -2, so our x-intercepts written as a coordinate pair would be (-1,0) and (-2,0).
Vertical Asymptotes Vertical asymptotes look like a vertical line on the graph. They often occur at a value of x where the function is undefined.
Sample Problem: (Show a graph)
f(x)=(x+2)/( x^2 +2x-8)=(x+2)/((x+4)(x-2))
This function cannot exist when x=-4 or when x=2 because that would make the denominator be zero, therefore this function has vertical asymptotes at x=-4 and x=2.
Horizontal Asymptotes Horizontal asymptotes explain what a function is doing as the x-values get really big or really small. Functions often approach a given value as x approaches positive or negative infinity. Functions could also be getting continually bigger (approaching positive infinity) or continually smaller (approaching negative infinity) as the x-values increase or decrease. The horizontal asymptote is the function value (y-value) that is being approached, or even hit. To find this, plug increasingly large (or small) numbers in for x and see what the function is doing.
Sample problem: (Show a graph)
Find the horizontal asymptote of the following function:
The horizontal asymptote tells me, roughly, where the graph will go when x is really, really big. So
I'll look at some very big values for x , some values of x very far from the origin:
x
increased by 2, so a x-value 2 below the one in the original function will yield the same point. Hence, the whole graph is slid to the left 2. What if the function had been changed to f(x)= (x-2)^2 Note: When changing y values “+” is up and “-“ is down. When changing x values “-“ is right and “+” is left.
Reflection A reflection is when a graph is flipped across the line, like a mirror image of itself. To flip (reflect) a graph across the y-axis we want to change all the x-values (x) so they produce the function value of their negatives (-x). It follows that to reflect across the y-axis we can just change the x-values to (-x). Note: Be careful to use your parentheses carefully so you are changing the x-value, not the y-value.
Sample: Write the equation of f(x)= x^2 +2 reflected about the y-axis. Insert (-x) for x so the new function becomes g(x)= (-x)^2 +2.
To reflect a graph across the x-axis, you want to change all y-values to their negative. To do so, put a negative throughout the whole function.
Sample: Write the equation of f(x)= x^2 +2 reflected about the x-axis. Our new function is the negative of our old function so g(x)=-f(x)= -x^2 -2. Warning: Be sure you apply the negative throughout the entire function.
Stretching and Compressing This is when a function becomes more compact or more spread out.
Pull out a graphing calculator again. What happens to the function f(x)=x when you multiply it by 3: f(x)=3x? You are moving through the y-values three times as fast, so the graph is compressed by 3 times the original. What if you multiplied by a fraction?
These are the basic ways to manipulate parent functions. You can combine any of these together. Feel free to play with a graphing calculator and see how doing different things affects the parent graph. Here are some problems you can work through.
Graph the following functions:
1.) f(x)=(x-3)^3 + 2 2.) f(x)=-2x^2 - 3.) f(x)=-2(-x)^2 -
