Graphing Using Asymptotes
Graph rational functions using asymptotes in Grade 9 algebra. Find vertical asymptotes where denominators equal zero and horizontal asymptotes from degree comparisons, then sketch the graph.
Key Concepts
Property Step 1: Identify and graph the vertical ($x=b$) and horizontal ($y=c$) asymptotes with dashed lines. Step 2: Make a table of values by choosing x values on both sides of the vertical asymptote. Step 3: Plot the points and connect them with smooth curves. Explanation Graphing a rational function is like an advanced connect the dots puzzle with invisible force fields. First, you draw your dashed line asymptotes—the 'no go' zones for your graph. Then, you plot a few key points on either side of the vertical line. Finally, connect the dots with smooth curves that get incredibly close to the asymptotes but never touch. Examples To graph $y = \frac{1}{x 2} + 1$: Asymptotes are $x=2$ and $y=1$. Plot points like $(1,0)$ and $(3,2)$ to draw the curves. To graph $y = \frac{ 2}{x+3}$: Asymptotes are $x= 3$ and $y=0$. Plot points like $( 4, 2)$ and $( 2, 2)$ to see the shape.
Common Questions
How do you find vertical asymptotes of a rational function?
Set the denominator equal to zero and solve for x. Each solution where the denominator equals zero is a vertical asymptote — the graph approaches but never crosses it.
How do you determine horizontal asymptotes?
Compare degrees: numerator degree < denominator → y = 0. Equal degrees → y = ratio of leading coefficients. Numerator degree > denominator → no horizontal asymptote.
How do asymptotes help you graph a rational function?
Draw asymptote lines first as guides, plot a few points in each region around them. The graph approaches but never crosses vertical asymptotes (unless the factor cancels).