Graphs of Equations Objective: To use many methods to sketch the graphs of equations

Graphs of Equations An equation in two variables represents the relationship between two quantities.

Graphs of Equations An equation in two variables represents the relationship between two quantities. A solution or solution point of an equation is an ordered pair that makes the equation true.

Graphs of Equations An equation in two variables represents the relationship between two quantities. A solution or solution point of an equation is an ordered pair that makes the equation true. ( 1,4) is a solution because if we substitute 1 for x and 4 for y, the equation is true.

Graphs of Equations An equation in two variables represents the relationship between two quantities. A solution or solution point of an equation is an ordered pair that makes the equation true. The graph of an equation is the set of all points that are solutions of the equation.

Determining Solutions Determine whether (2,13) and (-1,-3) are solutions to the equation y = 10x – 7

Determining Solutions Determine whether (2,13) and (-1,-3) are solutions to the equation y = 10x – 7 13 = 10(2) – 7; 13 = 20 – 7 True

Determining Solutions Determine whether (2,13) and (-1,-3) are solutions to the equation y = 10x – 7 13 = 10(2) – 7; 13 = 20 – 7 True -3 = 10(-1) – 7; -3 = -10 – 7 False

You Try Determine whether (2,13) and (-1,-3) are solutions to the equation y = 4x + 5

You Try Determine whether (2,13) and (-1,-3) are solutions to the equation y = 4x = 4(2) + 5; 13 = True -3 = 4(-1) + 5; -3 = False

Sketching Graphs of Equations There are two simple ways to graph an equation: 1)Plotting points 2)Using intercepts

Plotting points Graph y = 7 – 3x x y

Plotting points Graph y = 7 – 3x x y

Using intercepts Graph y = 7 – 3x The y-intercept is when x = 0. y = 7 (0, 7) The x-intercept is when y = 0. 0 = 7 – 3x 3x = 7 x = 2.33 (2.33, 0)

Absolute Value Graph y = |x – 1|

Absolute Value Graph y = |x – 1| The value that makes the function = 0 is the corner point of the graph. Let’s plot points around this zero to see what this looks like. x y

Symmetry We will look at three types of symmetry.

Symmetry We will look at three types of symmetry. 1) A graph is symmetric with respect to the x-axis if, whenever (x,y) is on the graph, (x,-y) is also on the graph.

Symmetry Algebraically, the graph of an equation is symmetric with respect to the x-axis if replacing y with –y yields the same equation.

Symmetry 2) A graph is symmetric with respect to the y-axis if, whenever (x,y) is on the graph, (-x,y) is also on the graph.

Symmetry Algebraically, the graph of an equation is symmetric with respect to the y-axis if replacing x with –x yields an equivalent equation.

Symmetry 3) A graph is symmetric with respect to the origin if, whenever (x,y) is on the graph, (-x,-y) is also on the graph.

Symmetry Algebraically, the graph of an equation is symmetric with respect to the origin if replacing x with –x and y with –y yields an equivalent equation.

Equations of Circles A circle centered at the origin has the form where r represents the radius of the circle.

Equations of Circles The standard form of the equation of a circle with center (h,k) and radius r is:

Equation of a Circle Find the equation of a circle with center (2, 4) and radius 4.

Equation of a Circle Find the equation of a circle with center (2, 4) and radius 4.

Equation of a Circle You Try: Find the equation of a circle with center (-1, 3) and radius 7.

Equation of a Circle You Try: Find the equation of a circle with center (-1, 3) and radius 7.

Equations of Circles The point (3,4) lies on a circle whose center is (-1,2). Find the equation of the circle.

Equations of Circles The point (3,4) lies on a circle whose center is (-1,2). Find the equation of the circle. If the center is (-1,2), we know that h = -1, k = 2.

Equations of Circles The point (3,4) lies on a circle whose center is (-1,2). Find the equation of the circle. If the center is (-1,2), we know this: We now need the radius. How will we find it?

Equations of Circles The point (3,4) lies on a circle whose center is (-1,2). Find the equation of the circle. If the center is (-1,2), we know this: We can use the distance formula.

Equations of Circles Use the distance formula (3,4), (-1,2)

Equations of Circles The point (3,4) lies on a circle whose center is (-1,2). Find the equation of the circle. If the center is (-1,2), we know this: We can do this another way. The point (3, 4) represents an order pair (x, y) that is on the circle. We can replace x with 3 and y with 4 to find the radius.

Equations of Circles The point (3,4) lies on a circle whose center is (-1,2). Find the equation of the circle. If the center is (-1,2), we know this:

Equations of Circles The point (3,4) lies on a circle whose center is (-1,2). Find the equation of the circle. If the center is (-1,2), we know this:

Homework Pages odd,49,51,57,59