Graphing Hyperbolas
![](/math/calculus/images/graphing_hyperbolas_img_1.png)
Remember the two patterns for hyperbolas:
![](/math/calculus/images/graphing_hyperbolas_img_2.png)
To graph a hyperbola....
1. Determine if it is horizontal or vertical. Find the center point, a, and b.
2. Graph the center point.
3. Use the a value to find the two vertices.
4. Use the b value to draw the guiding box and asymptotes.
5. Draw the hyperbola.
Examples:
![](/math/calculus/images/graphing_hyperbolas_img_3.png)
First, we know this is horizontal since the x is positive. That means the hyperbola will open to the left and right. The center point is (2, -1). a = 2, b = 3
The center is (2, -1). Let's plot that point:
![](/math/calculus/images/graphing_hyperbolas_img_4.png)
Next we will use our a value to find our two vertices. a = 2. Since the hyperbola is horizontal, we must count 2 spaces to the left and right of our center point. That will be the location of our vertices.
![](/math/calculus/images/graphing_hyperbolas_img_5.png)
Next, we'll use the b value to draw in a guiding box. b = 3, so we will count up 3 and down 3 from both vertices. This will give us the 4 corners of our guiding box.
![](/math/calculus/images/graphing_hyperbolas_img_6.png)
We can now draw our 2 asymptotes diagonally through the corners of the box:
![](/math/calculus/images/graphing_hyperbolas_img_7.png)
Finally, we draw in our hyperbola. Each half starts at the vertex and continues towards the asymptotes but never actually reaches them.
![](/math/calculus/images/graphing_hyperbolas_img_8.png)
The center point, guiding box, and asymptotes are not technically part of the answer, so a clean version of the graph would look like this:
![](/math/calculus/images/graphing_hyperbolas_img_9.png)
The center is (2, -1). Let's plot that point:
![](/math/calculus/images/graphing_hyperbolas_img_4.png)
Next we will use our a value to find our two vertices. a = 2. Since the hyperbola is horizontal, we must count 2 spaces to the left and right of our center point. That will be the location of our vertices.
![](/math/calculus/images/graphing_hyperbolas_img_5.png)
Next, we'll use the b value to draw in a guiding box. b = 3, so we will count up 3 and down 3 from both vertices. This will give us the 4 corners of our guiding box.
![](/math/calculus/images/graphing_hyperbolas_img_6.png)
We can now draw our 2 asymptotes diagonally through the corners of the box:
![](/math/calculus/images/graphing_hyperbolas_img_7.png)
Finally, we draw in our hyperbola. Each half starts at the vertex and continues towards the asymptotes but never actually reaches them.
![](/math/calculus/images/graphing_hyperbolas_img_8.png)
The center point, guiding box, and asymptotes are not technically part of the answer, so a clean version of the graph would look like this:
![](/math/calculus/images/graphing_hyperbolas_img_9.png)
![](/math/calculus/images/graphing_hyperbolas_img_10.png)
First, we know this is vertical since the y is positive. That means the hyperbola will open up and down. The center point is (-2, -3). a = 3, b = 1
The center is (-2, -3). Let's plot that point:
![](/math/calculus/images/graphing_hyperbolas_img_11.png)
Next we will use our a value to find our two vertices. a = 3. Since the hyperbola is vertical, we must count 3 spaces up and down from our center point. That will be the location of our vertices.
![](/math/calculus/images/graphing_hyperbolas_img_12.png)
Next, we'll use the b value to draw in a guiding box. b = 1, so we will count left 1 and right 1 from both vertices. This will give us the 4 corners of our guiding box.
![](/math/calculus/images/graphing_hyperbolas_img_13.png)
We can now draw our 2 asymptotes diagonally through the corners of the box:
![](/math/calculus/images/graphing_hyperbolas_img_14.png)
Finally, we draw in our hyperbola. Each half starts at the vertex and continues towards the asymptotes but never actually reaches them.
![](/math/calculus/images/graphing_hyperbolas_img_15.png)
Practice: Graph each hyperbola.The center is (-2, -3). Let's plot that point:
![](/math/calculus/images/graphing_hyperbolas_img_11.png)
Next we will use our a value to find our two vertices. a = 3. Since the hyperbola is vertical, we must count 3 spaces up and down from our center point. That will be the location of our vertices.
![](/math/calculus/images/graphing_hyperbolas_img_12.png)
Next, we'll use the b value to draw in a guiding box. b = 1, so we will count left 1 and right 1 from both vertices. This will give us the 4 corners of our guiding box.
![](/math/calculus/images/graphing_hyperbolas_img_13.png)
We can now draw our 2 asymptotes diagonally through the corners of the box:
![](/math/calculus/images/graphing_hyperbolas_img_14.png)
Finally, we draw in our hyperbola. Each half starts at the vertex and continues towards the asymptotes but never actually reaches them.
![](/math/calculus/images/graphing_hyperbolas_img_15.png)
![](/math/calculus/images/graphing_hyperbolas_img_16.png)
![](/math/calculus/images/graphing_hyperbolas_img_17.png)
![](/math/calculus/images/graphing_hyperbolas_img_18.png)
![](/math/calculus/images/graphing_hyperbolas_img_19.png)
![](/math/calculus/images/graphing_hyperbolas_img_20.png)
Answers:
![](/math/calculus/images/graphing_hyperbolas_img_21.jpg)
![](/math/calculus/images/graphing_hyperbolas_img_22.jpg)
![](/math/calculus/images/graphing_hyperbolas_img_23.jpg)
![](/math/calculus/images/graphing_hyperbolas_img_24.jpg)
![](/math/calculus/images/graphing_hyperbolas_img_25.jpg)
Related Links: Math Fractions Factors |