Hello Math Mavericks!
This cycle, we're transitioning from basic transformations to focusing on graphing and using quadratic functions to solve problems. Today, we're revisiting a fundamental skill: graphing functions using a table. This method is crucial for visualizing the relationship between inputs and outputs and understanding the behavior of different types of functions.
Understanding the Basics
The core idea behind graphing with a table is to choose several input values (x-values), substitute them into the function, and calculate the corresponding output values (y-values). These (x, y) pairs then become coordinates that we plot on a graph. By connecting these points, we can visualize the function.
Here's a step-by-step breakdown:
- Choose your x-values: Select a range of x-values that seem relevant to the function. For linear functions, just a few points suffice. For more complex functions, you might need more points to capture the curve accurately. When a domain is given, use those x-values!
- Calculate the y-values: Substitute each x-value into the function’s equation and solve for y. This gives you the corresponding y-value for each x-value.
- Create ordered pairs: Write each (x, y) pair as an ordered pair.
- Plot the points: Plot each ordered pair on the coordinate plane.
- Connect the points: Draw a smooth line or curve through the plotted points. If the domain is all real numbers, remember to add arrowheads to the "ends" of your graph to show that it continues infinitely.
Examples
Let's consider the function $f(x) = 2x + 1$. To graph it using a table, we can choose a few x-values, such as -2, -1, 0, 1, and 2.
| x | f(x) = 2x + 1 | (x, y) |
|---|---|---|
| -2 | $f(-2) = 2(-2) + 1 = -3$ | (-2, -3) |
| -1 | $f(-1) = 2(-1) + 1 = -1$ | (-1, -1) |
| 0 | $f(0) = 2(0) + 1 = 1$ | (0, 1) |
| 1 | $f(1) = 2(1) + 1 = 3$ | (1, 3) |
| 2 | $f(2) = 2(2) + 1 = 5$ | (2, 5) |
Now, we plot these points and connect them to form a straight line.
For a slightly more complex example, let's look at a quadratic function: $g(x) = x^2 - 2$
| x | g(x) = x^2 - 2 | (x, y) |
|---|---|---|
| -2 | $g(-2) = (-2)^2 - 2 = 2$ | (-2, 2) |
| -1 | $g(-1) = (-1)^2 - 2 = -1$ | (-1, -1) |
| 0 | $g(0) = (0)^2 - 2 = -2$ | (0, -2) |
| 1 | $g(1) = (1)^2 - 2 = -1$ | (1, -1) |
| 2 | $g(2) = (2)^2 - 2 = 2$ | (2, 2) |
Plotting these points reveals a parabola.
Domain Considerations
Remember that the domain of a function is the set of all possible input values (x-values). If a domain is specified, you should only use x-values within that domain to create your table. If no domain is specified, you can assume it's all real numbers unless the context suggests otherwise.
Today's Resources
- Class Notes - Review the concepts covered in class.
- Reteaching Worksheet - Extra practice to reinforce today's lesson.
- Reading Strategies Worksheet - Strategies to help you interpret and understand word problems.
Homework
Please complete the following worksheets to solidify your understanding:
Discussion Question
What is another strategy for graphing linear functions besides using a table? Share your thoughts in the comments below!
Keep up the great work, and I'll see you in the next class!