# Quadratic Equations and Functions

## Graph Quadratic Functions Using Transformations


### Graph Quadratic Functions of the form 

In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of  and ‘moving’ it according to information given in the function equation. We call this graphing quadratic functions using transformations.
In the first example, we will graph the quadratic function  by plotting points. Then we will see what effect adding a constant, k, to the equation will have on the graph of the new function 
The last example shows us that to graph a quadratic function of the form  we take the basic parabola graph of  and vertically shift it up  or shift it down .
This transformation is called a vertical shift.
Now that we have seen the effect of the constant, k, it is easy to graph functions of the form  We just start with the basic parabola of  and then shift it up or down.
It may be helpful to practice sketching  quickly. We know the values and can sketch the graph from there.
Once we know this parabola, it will be easy to apply the transformations. The next example will require a vertical shift.

### Graph Quadratic Functions of the form 


In the first example, we graphed the quadratic function  by plotting points and then saw the effect of adding a constant k to the function had on the resulting graph of the new function 
We will now explore the effect of subtracting a constant, h, from x has on the resulting graph of the new function 
The last example shows us that to graph a quadratic function of the form  we take the basic parabola graph of  and shift it left (h > 0) or shift it right (h < 0).
This transformation is called a horizontal shift.
Now that we have seen the effect of the constant, h, it is easy to graph functions of the form  We just start with the basic parabola of  and then shift it left or right.
The next example will require a horizontal shift.
Now that we know the effect of the constants h and k, we will graph a quadratic function of the form  by first drawing the basic parabola and then making a horizontal shift followed by a vertical shift. We could do the vertical shift followed by the horizontal shift, but most students prefer the horizontal shift followed by the vertical.

### Graph Quadratic Functions of the Form 

So far we graphed the quadratic function  and then saw the effect of including a constant h or k in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient a on the resulting graph of the new function 
If we graph these functions, we can see the effect of the constant a, assuming a > 0.
To graph a function with constant a it is easiest to choose a few points on  and multiply the y-values by a.

### Graph Quadratic Functions Using Transformations

We have learned how the constants a, h, and k in the functions,  and  affect their graphs. We can now put this together and graph quadratic functions  by first putting them into the form  by completing the square. This form is sometimes known as the vertex form or standard form.
We must be careful to both add and subtract the number to the SAME side of the function to complete the square. We cannot add the number to both sides as we did when we completed the square with quadratic equations.
When we complete the square in a function with a coefficient of x2 that is not one, we have to factor that coefficient from just the x-terms. We do not factor it from the constant term. It is often helpful to move the constant term a bit to the right to make it easier to focus only on the x-terms.
Once we get the constant we want to complete the square, we must remember to multiply it by that coefficient before we then subtract it.
Once we put the function into the  form, we can then use the transformations as we did in the last few problems. The next example will show us how to do this.
We list the steps to take to graph a quadratic function using transformations here.
Now that we have completed the square to put a quadratic function into  form, we can also use this technique to graph the function using its properties as in the previous section.
If we look back at the last few examples, we see that the vertex is related to the constants h and k.
In each case, the vertex is (h, k). Also the axis of symmetry is the line x = h.
We rewrite our steps for graphing a quadratic function using properties for when the function is in  form.

### Find a Quadratic Function from its Graph

So far we have started with a function and then found its graph.
Now we are going to reverse the process. Starting with the graph, we will find the function.

### Key Concepts

1. Graph a Quadratic Function of the form  Using a Vertical Shift


2. Graph a Quadratic Function of the form  Using a Horizontal Shift


3. Graph of a Quadratic Function of the form 


4. How to graph a quadratic function using transformations


5. Graph a quadratic function in the vertex form  using properties



### Practice Makes Perfect

Graph Quadratic Functions of the form 
In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, k, to the function has on the basic parabola.
In the following exercises, graph each function using a vertical shift.
Graph Quadratic Functions of the form 
In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, , inside the parentheses has
In the following exercises, graph each function using a horizontal shift.
In the following exercises, graph each function using transformations.
Graph Quadratic Functions of the form 
In the following exercises, graph each function.
Graph Quadratic Functions Using Transformations
In the following exercises, rewrite each function in the  form by completing the square.
In the following exercises, ⓐ rewrite each function in  form and ⓑ graph it by using transformations.
In the following exercises, ⓐ rewrite each function in  form and ⓑ graph it using properties.
Matching
In the following exercises, match the graphs to one of the following functions: ⓐ  ⓑ  ⓒ  ⓓ  ⓔ  ⓕ  ⓖ  ⓗ 
Find a Quadratic Function from its Graph
In the following exercises, write the quadratic function in  form whose graph is shown.

### Writing Exercise


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?