Graphing Section

7/31/2019 Graphing Section

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I. FUNCTIONS AND NOTATIONS

A. Relations and Functions

A function can be expressed in a table or a list form, but, a function can also be expressed in a

graph form. The domain of the function is plotted in the horizontal (x-axis), and the range is

plotted vertically (y axis). To graph a relation, we need to plot each given ordered pair. Here is an

example of a function:

Graph: *() () () ()+

B. Vertical Line Test

The definition of a function says that there can only be one range for a domain. Therefore, thegraph of a function cannot look like this:

In this graph, there are two or more corresponding y values for an

x value. This is called a vertical line test. We draw a straight line down

the graph and see the line passes through the graph at only one point each.

If it passes through only one point, it is a function.

C. Linear Functions

Linear functions are considered the simplest of all the types of functions. Linear functions are

the functions of the form, y = mx + b. These functions are straight lines where m is the slope and

b is the y-intercept. If m>0, the line rises, if m


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D. Absolute Value Function and Graph

Absolute value functions produce 2 solutions. When m 0, the functions are written as: () | | . In a graph of an absolute value function, there are two lines, a positiveand a negative. The two points meet up at one point called the vertex. The vertex is also where

either the maximum or the minimum is formed. The vertex can be found at the point ( ,c).

There are 3 steps in graphing an absolute value function:

1) Isolate the absolute value and find the vertex using ( , c).

2) Write an equation for where the absolute value is and one where the absolute value is .3) Graph the two equations found in step 2 on the correct domain.

Example:

Graph the absolute value function, | |Steps

1) Isolate the absolute value and find the vertex.

| | | |; m=1, b=1, c=4Vertex:( , c) = (-)

2) Write the 2 equations and solve

a) | |b) -| |

-

3) Graph the two equations found in step 2

on the correct domain.E. Translating, Stretching or Shrinking, Reflection

There are four things that affect the characteristics and the shapes of different graphs: translating,

stretching or shrinking, and reflection. These four things are represented as a, h, and k. The

parent function of an absolute value function is ||. Here is a chart for you!Translating, Stretching or Shrinking, Reflection

Parent Function ||y = |x + h The graph will move h units to the left

y = |x - h| The graph will move h units to the righty = |x| + k The graph will move k units up

y = |x| - k The graph will move k units down

a > 1 It is a stretch by the factor of a0 > a > 1 It is a shrink by the factor of a

y = -|x + h| Reflection over the x-axis

-8 -6 -4 -2 2 4 6

2

4

6

8

10

12

x

y

y=x+5y=-x+5

y=|x|

y=1/2 |x-2|+3


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F. Graphing Inequalities

An inequality has two variables and the graph is shaded to represent the inequality. The

disconnected dashed line (- - - - -) means > or or y , the shading is above the line and if the inequality is y < or y

, the shading is below the line.

Example:

Graph the inequality, Steps

1) Graph the line, 2) Pick a test point and plug the coordinates

into the inequality. If the point fulfills the

inequality, we shade the side of the point.

If I pick (-2, 4) 4 (-3) + 22 -33) Shade over the line because our coordinates

Over the line fulfilled the equation.

G. Absolute Values Inequality

Graphing absolute values inequality is the same as graphing a regular inequality but it is just that

you need to pick a test-point inside or outside the value and see if the points fulfill the inequality. If

the point fulfills, then you shade the area on the side of the line of where the test point is .

Example:

Graph the absolute value inequality, || Steps

1) Graph the absolute value inequality , || 2) Pick a test point and plug the coordinates

into the inequality. If the point fulfills the

inequality, we shade the side of the point.

If I pick (2, 6) 6 |()| 633) Shade between the lines because our coordinates

between the lines fulfilled the equation.

y (5/3)x + 2

y|3x| + 2


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II. GRAPHING LINEAR SYSTEMS

A. Graphing Systems of Equation

A solution of a system of equations is a set of values for the variables that makes all the equations true.

You can solve some linear system by graphing the equations. The solution is the point where the lines of graph

intersect.

Example:

Solve { by graphing.Steps

1) Graph the two equations

2) Find the intersecting point of the two lines.

3) The intersecting point is the solution = (1, 3)

Graphical Solutions of Linear Systems

One Solution, Independent No Solution, Dependent No Solution, Inconsistent

B. Graphing Systems of Inequalities

When we solve systems of inequalities by graphing, we must remember that the solutions include all

points on one side of the boundary line.

Example:

Solve the System of Inequalities, { Steps

1) Graph and shade both inequalities.

2) Where the two shadings intersect is your solution.

All the points in the intersected area is a solution of the inequalities.

x

y

x

y

x

y

x

y

x

y


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3) Pick a coordinate point in the shaded area and plug into the equation.

C. Linear Programming

We used Linear Programming to find the maximum or the minimum of an equation. The maximum or

the minimum value occurs at one or more vertices of the region that contains all points that satisfy the

constraints.

Example:

Find the values of x and y that maximize and minimize P if P = -5x + 4y and has the constraints Steps

1) Graph and Shade the Inequalities.

2) Plug the coordinates the objective equation

P = -5x + 4y to evaluate vertices

(1, 3): P = -5(1) + 4(3) = 7

(4, 1): P = -5(4) + 4(1) = -16

(5, 4): P = -5(5) + 4(4) = -9

3) The maximum 7 occurs at (1, 3). The minimum -16 occurs

at (4, 1)

f(x)=-(2/3)x+(11/3)

f(x)=3x-11

f(x)=(1/4)x+(11/4)


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III. GRAPH QUADRATIC EQUATIONS AND FUNCTIONS

A. Modeling Data WithQuadratic Functions

A graph of a quadratic function is called a Parabola. It has a shape of a bowl, with either a

minimum or a maximum which is the vertex of the parabola. A parabola has an axis of symmetry

which is mirror of the image divided from the vertex point. The y-value of the vertex is the

maximum or the minimum.

Minimum Maximum

B. Graphing Parabolas

In order to graph a quadratic function, you need at least three points. The standard form of a

quadratic function is y = ax2+ bx + c, a 0. When b = 0, the vertex of the parabola is (0, c) but

when b exists, we need to find the line of symmetry ry then further on calculate for thevertex by substituting the x value into the function.Example:

Sketch the curve, y = 2x28.

Steps

1) Graph the vertex, (0, c)

Vertex: (0, -8)

2) Make a table of values to find points on one side of

the graph.

x y

1 -6

2 0

3 10

y=-x^2

y=x^2

f(x)=2(x^2)-8


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3) Plot the corresponding points across the line of symmetry

and sketch the graph.

Sketch the curve, y = -x2 + 4x + 4.

Steps

1) Find the axis of symmetry and graph.

Axis of Symmetry: 22) Find the vertex.

When x = 2, y = -(22) + 4(2) + 4 = 8

Therefore, vertex = (2, 8)

3) Make a table of values to find points on one

side of the graph.

4) Sketch the curve

C. Translating Parabolas

As we discussed earlier, the standard form of a parabola is in the form f(x) = ax2 + bx + c. To

translate a parabola, we use the vertex form: y = a(xh)2 + k. The parent function of a parabola is y

= x2. Here are the different translations charts.

Vertical Stretch, Shrink, and Reflections

x y

0 -8

1 -6

2 0

3 10

Parent Function y = x2

Reflection in the x-axis y = -x2

Stretch (a>1) or shrink (0


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Shifting Right, Left, Up and Down

Parent Function y = a(xh)2

+ k

h < 0 Shifts Right

h > 0 Shifts Left

K < 0 Shifts Up

K > 0 Shifts Down

Example:

Graph the Equation, ( ) Steps

1) Find the vertex and the line of symmetry.

Vertex: (h, k) = (-2, -4)

L.O.S: x = h = -2

2) Make a table of values to find points on one

side of the graph.

x y

-2 -4-1 -3.3

1 2

4 20

3) Sketch the graph

IV. RADICAL FUNTIONS AND RADICAL EXPONENTS

A. Inverse Relations and Function

The inverse of a function contains the same values as the original function, except that the x

and y have been reversed. For example, if your function was made up with the

points*( ) ()()+, then the inverse of the function is*( ) ()()+. Basically,the inverse is the reflection over the line y = x.

f(x)=(2/3)((x+2)^2)-4

Line of Symmetry


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Example:

B.Graphing Square Roots and Radical Functions

In order to determine if an inverse of a function is a function or not, we can do the horizontal line

test. If the line intersects the graph of a function 2 or more times, then the inverse of the function

isnt a function. We can also restrict the domain of a function to make the inverse also a function.

The inverse of the function f(x) is denoted by f-1(x).

Radical Functions are in the form of Original Function Reflection in the x-axis Stretch (a>1) or shrink (0


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y=(1/x)

V. RATIONAL FUNCTIONS

A. The Reciprocal Function Family

The reciprocal function belongs to a group of functions with standard form , whosedomain is all real numbers but x h. The original reciprocal function is

and it looks like this:

d

There are 4 quadrants in the graphing plane.

When a>0, the function has branches in quadrants 1 and 3. When

a


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