Chapter 2.4 Graphs of Equations

Pindling College Algebra   by Example  Series

Key Concepts: Understand the properties of a graph and know how to graph any equation or function.

Skills to Learn:

1. Know how to tell the difference between an equation and a function

2. Know how to find the intercepts of a graph or function

3. Know how to graph any function

4. Know the basic equation of a circle and how to graph it

5. Know how to solve problems graphically

A Function - shows the relationship between one or more variables: Example is function that says, to find the value of y first square x and add 4. Note that for a function there are many possible values of y and x. For example: { (-1, 5), (0, 4), (1, 5), (10, 104), ... }

An equation - shows a function rule such as and possible state the domain or limiting values for that rule. Example is an equation that states only values for that are equal to 0. Note only two values of x, -2 and 2 satisfy the equation .

The y-intercept of a function is the value of the function when x = 0

The x-intercept(s) is the value of x when y = 0

Example 1. Find the intercepts of and graph its function. First solve for y = f(x) to get or The y-intercept (set x = 0 and solve for y) or (0, -4) A graph of the function would show this to be true. The x-intercept (set y = 0 and solve for x) , so or (-2, 0) and (2, 0)	To Graph the function Setup a table of values of x and y and plot each points: x (x, y) -5 (-5)2 - 4 = 21 (-5, 21) -1 (-1)2 - 4 = -3 (-1, -3) 0 (0)2 - 4 = -4 (0, -4) 1 (1)2 - 4 = -3 (1, -3) 5 (5)2 - 4 = 21 (5, 21) A rule of thumb is to select values of x from -10 through 0 to +10 and find corresponsing values of y to get a set of (x, y) points.
Graphs of	Parabola - Is a function with x2 as the highest power of the variable x. So is a parabola. Note that the graph is symmetrical about the y-axis (red line), so the line x = 0 is called the axis of symmetry. The lowest or highest point of a parabola is called the vertex. The vertex of the parabola shown is (0, 4). This graph is also call a quadratic function.

Example 2. Find the intercepts of and graph its function. The y-intercept (set x = 0 and solve for y) or (0, 1) A graph of the function would show this to be true. The x-intercept (set y = 0 and solve for x) , Remember that to remove the | | operator result in use of operator. So or (1, 0) and (3, 0)

To Graph the function Setup a table of values of x and y and plot each points: x (x, y) -5 7 - 1 = 3 (-5, 3) -1 3 - 1 = 2 (-1, 2) 0 2 - 1 = 1 (0, 1) 1 1 - 1 = 0 (1, 0) 5 3 - 1 = 2 (5, 2) A rule of thumb is to select values of x from -10 through 0 to +10 and find corresponsing values of y to get a set of (x, y) points.

Graphs of

Absolute value - Is a function with operator | | where any operation with the | | are assign a positive value. Example |-2| = 2, |3-7| = 4 Note that the graph is symmetrical about the vertical line where the slope of the line change from positive to negative (red line), so the line x = 2 is called the axis of symmetry. The lowest or highest point of a parabola is called the vertex. The vertex of the parabola shown is (0, 4).

Example 3. Find the intercepts of and graph it. The y-intercept (set x = 0 and solve for y) First solve for y = f(x) to get or (0, -4) and (0, -4) A graph of the function would show this to be true. The x-intercept (set y = 0 and solve for x) , Remember that to remove the operator one must square both sides. So or (-4, 0) and (4, 0)	To Graph the function Setup a table of values of x and y and plot each points: x (x, y) (x, y) -5 U U -4 (-4, 0) (-4, 0) -1 (-1, 3.9) (-1, -3.9) 0   (0, 4) (0, -4) 1   (1, 3.9) (1, -3.9) 4 (4, 0) (4, 0) 5   U U U means undefined or no solution
Graphs of or and The equation for this circle is: or same as orginal equation.	Circle with a center (h, k) and radius, r is defined by this equation: When (h, k) is the origin or (0, 0) The equation is Example a circle with center (4, -5) and radius 12 is given by the equation: or

Find the equation ofa circle with center (5, 0) and a point on the circle of (1, 0): A sketch of the circle with center (5, 0) shows that the radius is 4. Equation of a circle is , where (h, k) is the x-y coordinates of the center and r is the radius. So the equation is:

Example 4. Find the intercepts of and graph it. The y-intercept (set x = 0 and solve for y) or (0, 3) A graph of the function would show this to be true. The x-intercept (set y = 0 and solve for x) , Remember that to remove the operator one must square both sides. So or (9, 0)	To Graph the function Setup a table of values of x and y and plot each points: x (x, y) -10   (-10, 4.36) -5   (-5, 3.74) -1   (-1, 3.16) 0   (0, 3) 1   (1, 2.83) 5   (5, 2) 9   (9, 0) 10   U U means undefined or no solution
Graphs of	Square root functions various depending on many factors; see if you could find the intercepts and graph See graphical solution below:

Example 5. Find intercepts and graph First solve for y = f(x) to get So graph Intercepts are: (0, 5.2) and (-13, 0)

Example 6. Find intercepts and graph Intercepts (-2, 0) and (2, 0) - No y-intercept

Look at these functions and their graphs

Graphical Solutions to problems

Example 7 Find the points of intersection of these two functions Note points of intersection from graphs.

Example 8. Find the point (x, y) where these two lines are equal: Try and find a solution that is closed to the true soltion.

-2x = 4x - 4, -6x = -4, x = 2/3 = 0.67

So y = -2 (0.67) = 1.34