Graphs of Equations in Two Variables Including Graphs of Functions (2.1, 2.2)

POD Show that (9, 1) is a solution of (or satisfies) the equation 2x – 3y = 15. What does it mean to be a solution? How many solutions would this equation have?

Equations in Two Variables Give an example of an equation with two variables. Don’t make it complicated. What values do you find for your equations? How do we graph them? (Think Rene Descartes, )

For 2x – 3y = 15 Plug in x = 0, x =2, x = 4, and x =6. What do you get for y in each case? What do these numbers represent?

For 2x – 3y = 15 Plug in x = 0, x =2, x = 4, and x =6. (0, -5), (2, -11/3), (4, -7/3), (6,-1). These are solutions to the equation– they are x and y values that make the equation true.

For 2x – 3y = 15 Graph (0, -5), (2, -11/3), (4, -7/3), (6,-1). As neatly as you can, connect the dots. What is this? What does this line represent? This line is made up of all the points that are the x-y values that make the equation true. It’s a picture of the solution set!

For 2x – 3y = 15 The linear equation above is written in standard form. Why is it linear? How would you rewrite it into slope-intercept form? (Okay, what is slope-intercept form?) This is a snap with CAS.

For 2x – 3y = 15 y = 2/3 x -5 Two advantages to this form: 1. You can identify the slope and y-intercept quickly, so it’s easy to graph by hand. 2. It’s easy to type into the graphing calculator. Let’s do that.

For 2x – 3y = 15 y = 2/3 x -5 What do you see? The intercepts and slope. How could we determine the x- and y-intercepts? The slope?

Try y = x 2 Make a table of several x values and their corresponding y values. Make sure you include some negative x values. Graph those points and connect the dots. Graph on calculators to check your work. The previous graph was a line. What is the shape of this graph?

Try y = x 2 This is a parabola. Let’s talk about graphs in general.

Relation vs. Function One important consideration for graphs is whether or not they are functions. In a function, there is only one y value for any x value. In other words, the graph passes the vertical line test (the VLT). We’ve looked at two functions. Sketch a graph of a function that is only a relation. Show us.

Domain and Range All graphs, whether function or not, have a domain and range. Domain is all possible x values for the graph. Range is all possible y values for the graph.

Domain and Range Sometimes they are restricted and sometimes they aren’t. In our line, x could be any real number, so the domain is R (all real numbers). In the line, the range is also R.

Domain and Range In our parabola, x can be any real number, so again the domain is R. But notice how the y values are always 0 or positive. The range is R 0. We could also say the range is all non- negative numbers.

Domain and Range Sometimes they are restricted and sometimes they aren’t. What is the domain and range of this graph? Is it a function? Why or why not?