THE PYTHAGOREAN THEOREM NONLINEAR GRAPHS

This unit begins with the Pythagorean Theorem and how it applies when calculating the lengths of the legs and the hypotenuse of a right triangle. The Pythagorean Theorem is a very useful formula to know when determining various right-angle measurements. The second part of this unit is about nonlinear functions and their graphs. The graphs of the equations will be nonlinear (not a straight line). Some examples are graphs of quadratic functions and graphs of inverse relationships.

Pythagorean Theorem

Graphing Quadratic Functions

Inverse Variation

Linear and Non-Linear Graphs

The Pythagorean Theorem

The Pythagorean Theorem is used to find the lengths of the sides of right triangles.

The Pythagorean Theorem

abc222+ =

a and b are the lengths of the legs and c is the length of the hypotenuse (the side opposite the right angle).

A c hypotenuse b leg B C a leg

To use the Pythagorean Theorem in finding the missing measures, follow the examples below.

Example 1: Find the length of side x (the hypotenuse of the right triangle) .

What is the value of x?

abc222+= 1222+ 9 = x 2

x 2 9 144+ 25 = x 169 = x2 12 169 = x 2 13 = x

Example 2: Find the length of side y (the hypotenuse of the right triangle).

What is the value of y?

abc222+= 5322+=y 2 5 3 25+= 9 y 2 34 = y 2 y

34 = y 2

y ≈ 5.83 (rounded to the nearest hundredth)

Example 3: Find the length of side d, the diagonal of the square.

abc222+= 4422+=d 2

2 d 16+= 16 d 4 d 2 = 32 4 d 2 = 32

d ≈ 5.66 (rounded to the nearest hundredth)

The Pythagorean Theorem can also be used to find the measure of the legs of a right triangle.

Example 4: Find the length of side a (a leg of the right triangle).

abc222+= A a 22+=10 26 2 26 a 2 +=100 676 10 a 2 = 576

B 2 C a a = 576 a = 24

Example 5: Find the length of side b (a leg of the right triangle).

abc222+= A 4222+=b 60 2 1764+=b2 3600 60 b 1764 −1764 +b2 = 3600−1764 2 C B b =1836 42 b2 = 1836

b ≈ 42.85 (rounded to nearest hundredth)

Application

Example 6: You are traveling on your bicycle due north at 20 miles per hour. You friend, who left at the same time traveling due east at 15 miles per hour, called you an hour later on the cell phone to tell you he had a flat tire. How far must you travel to meet your friend? (Assume you can travel directly to your friend’s location.)

North at 20 mph Distance you must travel to meet your 20 miles friend 15222+= 20 d 225+ 400 = d 2

625 = d 2

East 625 = d 2 at 15 mph 15miles 25 = d

The distance you must travel to reach your friend is 25 miles. Graphing Quadratic Functions

Quadratic Function

A quadratic function is a function of the form yaxbxc= 2 ++ where a, b, and c are real numbers and a ≠ 0.

The graph of a quadratic function is a curve known as a parabola and is shown below. y = x2

ƒ The lowest point on this parabola is the minimum value of the function.

ƒ The point (0, 0) on this graph and is called the vertex of the parabola.

ƒ In a parabola there is a vertical line called the axis of symmetry drawn through the vertex, that reflects the parabola across the line of x = h or in other words splits the parabola into two equal parts. In the case below the axis of symmetry would be the y-axis or x = 0.

If the coefficient of x-squared is a negative number, the parabola opens downward, and therefore has a maximum value at the vertex.

y =−x2

ƒ The equation of this graph can be written as yx=−(1)2 . The coefficient of x-square is –1.

ƒ The highest point on this parabola is the maximum value of the function.

ƒ The point (0, 0) on this graph and is also called the vertex of the parabola.

A table of values is very useful when examining the graph a quadratic equation.

Example 1: Graph yx= 2 + 2

a) Make a table of values using positive and negative x-values.

x x2 + 2 y

–2 (2)−+2 2 6 –1 (1)−+2 2 3 0 (0)2 + 2 2 1 (1)2 + 2 3 2 (2)2 + 2 6

b) Graph the points from the table on a coordinate plane.

c) Connect the points using a curve (remember that parabolas are curved lines).

*Notice that the vertex of this parabola is located at (0, 2) as opposed to (0, 0) as shown in the first part of the unit. Let’s take a look at both functions and decide why this happened.

2 a) y = x b) yx= 2 + 2

Compare both equations and notice: ƒ In the first graph’s equation yx= 2 , there is no constant term. ƒ In the second graph’s equation yx= 2 + 2, there is a constant of + 2. ƒ The graph of the second equation has moved up the y-axis 2 units.

*It can be concluded that a constant moves the graph vertically along the y-axis. Let’s use yx= 2 as the parent function (original function) with a vertex at (0, 0) and compare it with yx= 2 − 3. What will happen in this case?

yx= 2 yx= 2 − 3 table of values 2 for yx=−3 x y –2 1 –1 –2 0 –3 1 –2 2 1

*Due to the pixels in the graphing calculator it may look like the second equation passes through (2, 0) and (–2, 0). This is not the case and at this point we are only concerned about what the constant does to the graph.

By examining the two equations you can see that by subtracting 3 the graph has moved down 3 units from the origin on the y-axis.

Example 2: Graph yx=+(1)2

Make a table of values and graph the points.

x y –2 1 –1 0 0 1 1 4

Notice that in this example, the number of points on each side of the vertex is not the same. More values for x can be substituted to determine how wide the parabola gets on the left.

*Notice that when the constant term is in a quantity with the variable, the vertex moved horizontally along the x-axis. In the case above it moved one unit to the left. What would happen to the vertex of the equation yx=−(2)2 ?

If you answered that the vertex would move 2 units to the right, you are correct.

Inverse Variation

An inverse variation is a function that is defined by an equation in the following form: xyk= where k is a nonzero real-number constant.

*There are many situations in which one quantity varies indirectly as another:

ƒ as the rate increases, the time decreases when traveling a set distance.

ƒ the volume of a gas in a container decreases as the pressure increases and the temperature remains constant.

Consider the following expressions:

xy= k x y k = Divide both sides by x . x x k y = x 11k yk=⋅ Another way to write is k ⋅ . x xx 1 Thus, yx is directly proportional to the multiplicative inverse of , . x

y varies inversely to x

Example 1: Examine the following function: xyk= 30 (= 30) . What are some values for y that correspond to x?

The given function is graphed below. Study the graph carefully. Notice that as x increases, y decreases.

30 1 xy=→=→=⋅30 y y 30 xx

For example: When x is 1, y is 30…(1, 30) . When x increases to 2, y decreases to 15…(2, 15). When x increases to 10, y decreases to 3…(3, 10). When x increases to 20, y decreases to 1 1/2…(20, 1.5).

y

x

*Note: the graph is nonlinear.

Example 2: If x varies inversely to y and x = 3, find y if the constant of variation (k) equals 24.

xyk==24 24 3yx== 24 Substitute 3. y = 8 Solve for y.

When x = 3, y = 8.

Example 3: Find the constant of variation when x varies inversely to y with x = 5 and y = 20.

xyk= 4(5)=== 20 Substitute xy4, 5. 20= kk Solve for .

When x = 4 and y = 5, k = 20.

Example 4: The members of the Drama Club are going to paint the scenery for their play. They figured that it would take 4 members 12 days to paint the scenery. If they needed the scenery completed in 8 days, and they work at the same rate, how many members do they need to help paint?

This is an inverse variation problem. As the number of members increase in helping out with the painting, the time it takes to complete the job will decrease.

Let x = # of members and y = # of days.

First, find the constant of variation when x varies inversely to y with x = 4 and y = 12.

xyk= 4(12)=== 48 Substitute xy4, 12. 48= kk Solve for .

When x = 4 and y = 12, k = 48. Now find how many members are needed to complete the job in 8 days. xyk= xyk(8)=== 48 Substitute 8, 48. xx= 6Solve for .

To complete the job in 8 days, it will take 6 members all working at the same rate.

Linear and Nonlinear Equations

Now that we have graphed equations, let’s see if we can reverse the process to predict what the equation might look like if we only “see the graph or the table of values”. In this section we will examine both a linear graph and a quadratic graph.

Example 1: Study the table of values and look for a relationship between the x-values and the y-values.

x y 1 2 2 4 3 6 4 8

The y-value is equal to the x-value multiplied by two. We can write an equation for the relationship, y = 2x. The graphed points from the table suggest a linear equation. Since the graph is a straight line, we can assume the equation is linear or x is to the first degree.

y = 2x

The equation for the table of values is y = 2x and the graph is linear.

Example 2: Study the table and look for a relationship between the x-values and the y-values.

x y –3 9 –2 4 –1 1 0 0 1 3 2 4 3 9

Notice that the negative values are positive y-values. Therefore, it can be concluded that the x-term is squared and a parabola is the result. The x-term is to the second degree or squared.

y = x2

The equation for the table of values is y = x2. This graph is nonlinear because it forms a curved line and is a parabola, not a straight line.

The next graph represents an inverse variation relationship, as x increases, y decreases.

Example 3: Graphxy = 30.

x y 1 30 2 15 3 10 5 6 6 5 10 3 15 2 30 1

Equations that represent inverse variation are curved graphs; thus, they are nonlinear equations.

Now, let’s examine one more type of graph. What is the shape of a graph when the x-values are raised to the third power?

Example 4: What is the corresponding y-values of the equation yx= 3 when x = –2, when x = –1, when x = 0, and when x = 3? Graph these points.

Solution:

When xy = −2, =− (2)3 → y =−−− (2)(2)(2) → y =− 8 When xy = −=−→1, ( 1)3 y =−−−→ ( 1)( 1)( 1) y =− 1

When xy = 0, =→= (0)3 y (0)(0)(0) →= y 0 When xy = 3, =→= (3)3 y (3)(3)(3) →= y 27

The graph of these points and many others that make up the curve of y = x3 appear as show in the graph below.

y = x3

*This graph was created using a graphing calculator.

Equations that are built on the basic graph of y = x3 are nonlinear.