# Precalculus : Graphing Functions

## Example Questions

### Example Question #2 : Determine The Equation Of A Linear Function

Find the equation of the line with slope  that passes through the point .

Explanation:

Since our slope is , we can plug it into  right away giving .

To solve for , we plug our given point  in for  and  giving .

This will simplify to , or  after subtracting the fraction.

Hence our answer is  after plugging our values for  and  in.

### Example Question #3 : Determine The Equation Of A Linear Function

Suppose Bob has 4 candies.  He then earns candies at a rate of 14 candies per week.  Which of the following formulas is the most reasonable rate to know how many candies Bob has on any given day?

Explanation:

Bob starts out with 4 candies.  Write the equation.

Every week, he earns 14 candies.  Every week has a total of 7 days.  Divide the amount of candies by the number of days to determine the number of candies Bob earn in a day.

Bob then earns 2 candies per day.  Let  represent the number of candies per day.  Finish the incomplete equation.

### Example Question #4 : Determine The Equation Of A Linear Function

After one hour of working, Billy has three dollars.  At the end of eight hours, Billy has fifty dollars.  Choose the most correct linear function representing this scenario if  represents time, and  represents dollars.

Explanation:

To solve for the equation of this problem, we will need 2 points.  Let the point be defined as , where  is time in hours and  represents the dollars Billy has earned.

Billy starts with three dollars in the first hour.  Write the point.

At the end of eight hours, Billy has fifty dollars.  Write the second point.

Write the slope-intercept formula.

Write slope formula.

Plug a point and the given slope to the slope-intercept formula to solve for the y-intercept.

Substitute this and the slope back into the slope-intercept equation.

Since Billy's earning is dependent on the time he works, rewrite the equation so that time is the independent variable, and amount earned is the dependent.

### Example Question #5 : Determine The Equation Of A Linear Function

Determine the equation of a line that passes through the points  and .

Explanation:

Linear functions follow the form , where m is the slope and b is the y intercept. We can determine the equation of a linear function when we have the slope and a y intercept which is a starting point for drawing our line. If we need the equation of a line that passes through two points we use the point slope equation:

where the variables with numbers next to them correspond to places where you input the x and y from a single (same) coordinate pair.

Since we have two points, we can compute the slope of a line that passes between them.

.

Now use the point slope formula:

distribute the right side

Take a look at the coordinate pairs. Was it necessary to to use the point slope formula? The answer is no, since the coordinate pair (0,5) is already a y intercept. So all that was necessary was to compute the slope.

### Example Question #6 : Determine The Equation Of A Linear Function

What equation is perpendicular to  and passes throgh ?

Explanation:

First find the reciprocal of the slope of the given function.

The perpendicular function is:

Now we must find the constant, , by using the given point that the perpendicular crosses.

solve for :

### Example Question #7 : Determine The Equation Of A Linear Function

What equation is perpendicular to  and passes throgh ?

Explanation:

First find the reciprocal of the slope of the given function.

The perpendicular function is:

Now we must find the constant, , by using the given point that the perpendicular crosses.

solve for :

### Example Question #8 : Determine The Equation Of A Linear Function

What equation is perpendicular to and passes throgh ?

Explanation:

First find the reciprocal of the slope of the given function.

The perpendicular function is:

Now we must find the constant, , by using the given point that the perpendicular crosses.

solve for :

### Example Question #1 : Graph A Quadratic Function Using Intercepts, Vertex, And Axis Of Symmetry

Find the vertex, roots, and the  value that the line of symmetry falls on of the function .

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on x=.5.

vertex , the roots  and , and the axis of symmetry would fall on .

Explanation:

All quadratic functions have a vertex and many cross the x axis at points called zeros or roots. If we know the vertex and its zeros, quadratic functions become very easy to draw since the vertex is also a line of symmetry (the zeros are equidistant from the vertex on either side).

Factor the equation  to get  and . Thus, the roots are 3 and -2.

The vertex can be found by using .

simplify

.

The axis of symmetry is halfway between the two roots, or simply the x coordinate of the vertex. So the axis of symmetry lies on x=1/2. To graph, draw a point at the coordinate pair of the vertex. Then draw points on the x axis at the roots, and finally, trace upwards from the vertex through the roots with a gentle curve.

### Example Question #2 : Graph A Quadratic Function Using Intercepts, Vertex, And Axis Of Symmetry

Which of the following functions matches the provided parabolic graph?

Explanation:

Finding the vertex, intercept and axis of symmetry are crucial to finding the function that corresponds to the graph:

The vertex form of a quadratic function is written as:

and the coordinates for the vertex are:

Looking at the graph and the position of the axis of symmetry, the vertex is positioned at , leaving us with an equation so far of:

While we don't know a right away,  is the only option that really works. The y-intercept is at  and we can plug that into the formula to confirm that this is the correct function:

### Example Question #3 : Graph A Quadratic Function Using Intercepts, Vertex, And Axis Of Symmetry

Which of the following is an equation for the parabola represented in the graph below?

Explanation:

Immediately we can tell that the equation has a negative coefficient, because the parabola opens downward, forming an umbrella shape. Based upon the information given in the figure, we can use the intercepts, axis of symmetry, and the vertex to identify the equation of the parabola. Let's observe the vertex form of a parabola written as the following:

In this equation,  is the vertex of the parabola, and  determines whether the parabola opens upwards or downwards. The axis of symmetry is at  and the vertex is located at , which we can plug into the following function:

We know that  is negative because of the position of the parabola.