Question

y-intercept:
there is a y-intercept at (0,0).
there is no y-intercept.

end behavior:
there is a horizontal asymptote at .
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graph the function on your paper packet! dont check with calculator

my graph looks like:

Explanation:

🆕 New Concept Discovered: End Behavior of Rational Functions
How a graph behaves at its far left and far right ends.

Step 1: Identify the function type from the graph

The graph shown is a rational function with vertical asymptotes (indicated by the dashed red lines) and a horizontal asymptote.

To determine the end behavior, we look at what happens to the function's output values (\(y\)-values) as the input values (\(x\)-values) head towards positive infinity (\(\infty\)) and negative infinity (\(-\infty\)).

Step 2: Analyze the far-left and far-right behavior

Looking at the blue curves on the graph:

• As we follow the graph to the far left (\(x \to -\infty\)), the curve flattens out and approaches the horizontal line \(y = 0\) (the x-axis).
• As we follow the graph to the far right (\(x \to \infty\)), the curve also flattens out and approaches the horizontal line \(y = 0\).

Because the graph approaches a specific constant value on both ends rather than shooting up to \(\infty\) or down to \(-\infty\), the end behavior is defined by a horizontal asymptote.

Step 3: Determine the horizontal asymptote equation

The horizontal line that the graph approaches on both the far left and the far right is the x-axis, which has the equation:

$$y = 0$$

Therefore, the correct option under "End Behavior" is:
"There is a horizontal asymptote at" with the value \(0\) (or \(y = 0\) depending on the input format, typically just the number \(0\) or the equation \(y=0\)).

Answer:

Select the first option:

• There is a horizontal asymptote at 0 (or y = 0)