4-1 Notes Graphing Quadratics in Vertex form
Let's start by trying a few things out.
1. Simplify the following expressions:
a. LaTeX: 3\left(x+5\right)+2
b. LaTeX: 6\left(x-3\right)+x-4
c. LaTeX: x\left(x+2\right)-6x
d. LaTeX: 3x\left(2x-5\right)+8x
e. LaTeX: \left(3x\right)^2+x^2
f. LaTeX: 3x^2-5x+2x^2+8
If you simplified these correctly, they should look like this.
If your answers differ from mine, look through and see where your work (or maybe mine) went wrong. If there's an issue with my work, send me a message and I'll fix it.
If you notice, most of these equations have another term we haven't graphed yet. It's the LaTeX: x^2term. We'll be looking at this term and hopefully be able to identify key features and be able to graph this function by the end of the day.
Let's work on simplifying a few more expressions:
2. Simplify the following.
a. LaTeX: x\left(x+4\right)
b. LaTeX: 3\left(x+4\right)
c. LaTeX: \left(x+3\right)\left(x+4\right)
In a, the simplified expression should be: LaTeX: x^2+4xand for b, the expression should be: LaTeX: 3x+12. So when we solve c, since it's asking us to distribute LaTeX: x+3into LaTeX: x+4, all we need to do is add the previous two statements together to get: LaTeX: x^2+4x+3x+12
We can then combine terms to get: LaTeX: x^2+7x+12
The same process applies if we need to simplify this expression here:
3. LaTeX: \left(x-5\right)\left(2x+3\right)
First start by distributing the x to the 2x and 3 you get: LaTeX: 2x^2+3x
Then distribute the -5 to the 2x and 3. You get: LaTeX: -10x-15(don't forget the negatives)
Combine like terms: 3x and -10x to get -7x
So your simplified equation is: LaTeX: 2x^2-7x-15
Let's practice a few of these.
3.
a. LaTeX: \left(3x-5\right)\left(2x+1\right)
b. LaTeX: \left(x-4\right)\left(x+5\right)
c. LaTeX: \left(x+9\right)\left(x+2\right)
d. LaTeX: \left(4x-1\right)\left(4x+1\right)
Once you have completed those, check your work here.
Let's look at another few examples:
4. LaTeX: \left(x+5\right)^2
while this one doesn't look like the ones we did on 3, it has the same principle. Be careful! A lot of students get this wrong because they think it's easy: Just square both terms: LaTeX: x^2+25. WRONG!!!!! DON'T DO THAT. Why is it not that way?
Remember: Raising something to the second power, means to times it by itself. But we need to realize: we need to times x+5 by itself. So, whenever you see something like this, we need to think: LaTeX: \left(x+5\right)^2=\:\left(x+5\right)\left(x+5\right)and we can simplify from there. When we distribute, we get: LaTeX: x^2+5x+5x+25.
Then we can combine the like terms and get: LaTeX: x^2+10x+25
Let's practice a few of these.
5.
a. LaTeX: \left(x-3\right)^2
b. LaTeX: \left(x-5\right)^2
c. LaTeX: \left(x-1\right)^2
d. LaTeX: \left(x+8\right)^2
Once you have simplified those, check your answers here.
We can now take this practice one step further.
6. Simplify the following expression: LaTeX: \left(x+5\right)^2-4
For this expression, we will simplify the first part - like we did above - and that will give us: LaTeX: x^2+10x+25\:-4. Then combine the numbers to give us: LaTeX: x^2+10x+21
This last question illustrates the two forms of a quadratic equation that we can have. When graphing quadratic equations, we can graph an equation like: LaTeX: f\left(x\right)=\left(x+3\right)^2-5or an equation that gives the same graph: LaTeX: f\left(x\right)=x^2+6x+4.
Our task today is to learn how to graph the first type: LaTeX: f\left(x\right)=\left(x+3\right)^2-5. We call this type vertex form of a quadratic. Let's first figure out what the parent function of this graph is.
Parent function: LaTeX: f\left(x\right)=x^2
Now create a table of values and plot those points on a graph:
x f(x)
-3
-2
-1
0
1
2
3
Your table of values and what the graph should look like is here.
We call this shape a parabola. That's a good word to remember. Write it down.
What is the rate of change between each value? Find the slope between each point using: LaTeX: \frac{y_2-y_1}{x_2-x_1}
It's a new rate of change we haven't discussed before. Our rates are: -5, -3, -1, 1, 3, 5. In fact, if we expand our graph, what do you think the next few rates are?
We can understand what the rates are by finding the difference in all our perfect squares. What are the first 8 perfect squares? LaTeX: 0^2=0,\:1^2=1,\:2^2=4,\:3^2=9and so forth.
Just looking at the results, we have: 0, 1, 4, 9, 16, 25, 36, 49, 64... What is the rate of change between these values? 1, 3, 5, 7, 9, 11, etc. Let's keep these numbers in mind whenever we are graphing quadratic functions.
Let's do another table of values and graph them.
LaTeX: g\left(x\right)=2x^2
x g(x)
-3
-2
-1
0
1
2
3
Plot these points on a graph. How is this graph related to the parent function: LaTeX: f\left(x\right)=x^2
The table and graph for this function is found here.
What is the rate of change between these values? How are those related to the rates of change from the parent function?
This graph has a rate of change of 2, 6, 10 on the right side of the vertex (the point at the bottom of the graph). If we refer back to the rate of change between our squares, how are these related to 2, 6, 10? We multiply the 1, 3, 5, 7, 9.... by the leading coefficient (the 2 in front of the equation) and that gives us: 2, 6, 10, 14, 18. The number in front determines how much you'll multiply the numbers 1, 3, 5, 7, 9... by.
So the graph of LaTeX: y=3x^2has rates of.... 3, 9, 15, 21.
What would the rates of LaTeX: y=\frac{1}{2}x^2be?
LaTeX: \frac{1}{2},\:\frac{3}{2},\:\frac{5}{2},\:\frac{7}{2}
So if we look at the graph of LaTeX: y=\frac{1}{2}x^2, the vertex is at (0, 0) and every unit we move to the right, our y-values increase by LaTeX: \frac{1}{2},\:\frac{3}{2},\:\frac{5}{2},\:\frac{7}{2}.
25.png
If the leading coefficient is a negative, let's say LaTeX: y=-3x^2, the graph would be decreasing by -3, -9, -15 etc.
26.png
So the only thing left we have to figure out is if we need to graph an equation like:
LaTeX: h\left(x\right)=-2\left(x-3\right)^2+5
From looking at this graph, the scale is going to be -2, -6, -10, but where is the vertex? What are the transformations in this graph? How is the parent function transformed?
Right 3, up 5. So the vertex of this parabola is at (3, 5) and the next points are going to be down 2, over 1. Then down 6, over 1. Then down 10, over 1.
27.png
Let's look at one more:
LaTeX: j\left(x\right)=3\left(x-5\right)^2-8
The vertex is where? (5, -8) (Remember, if it's with the x, think opposite)
28.png
If you have more questions, or need another explanation, I recorded this video last year that might help you out a little.
If you have questions, feel free to post them on the discussion board and I'll try to answer them as quickly as I can. (or if you know the answers to any questions on the discussion board, you can answer them and help your classmates).
1. Simplify the following expressions:
a. LaTeX: 3\left(x+5\right)+2
b. LaTeX: 6\left(x-3\right)+x-4
c. LaTeX: x\left(x+2\right)-6x
d. LaTeX: 3x\left(2x-5\right)+8x
e. LaTeX: \left(3x\right)^2+x^2
f. LaTeX: 3x^2-5x+2x^2+8
If you simplified these correctly, they should look like this.
If your answers differ from mine, look through and see where your work (or maybe mine) went wrong. If there's an issue with my work, send me a message and I'll fix it.
If you notice, most of these equations have another term we haven't graphed yet. It's the LaTeX: x^2term. We'll be looking at this term and hopefully be able to identify key features and be able to graph this function by the end of the day.
Let's work on simplifying a few more expressions:
2. Simplify the following.
a. LaTeX: x\left(x+4\right)
b. LaTeX: 3\left(x+4\right)
c. LaTeX: \left(x+3\right)\left(x+4\right)
In a, the simplified expression should be: LaTeX: x^2+4xand for b, the expression should be: LaTeX: 3x+12. So when we solve c, since it's asking us to distribute LaTeX: x+3into LaTeX: x+4, all we need to do is add the previous two statements together to get: LaTeX: x^2+4x+3x+12
We can then combine terms to get: LaTeX: x^2+7x+12
The same process applies if we need to simplify this expression here:
3. LaTeX: \left(x-5\right)\left(2x+3\right)
First start by distributing the x to the 2x and 3 you get: LaTeX: 2x^2+3x
Then distribute the -5 to the 2x and 3. You get: LaTeX: -10x-15(don't forget the negatives)
Combine like terms: 3x and -10x to get -7x
So your simplified equation is: LaTeX: 2x^2-7x-15
Let's practice a few of these.
3.
a. LaTeX: \left(3x-5\right)\left(2x+1\right)
b. LaTeX: \left(x-4\right)\left(x+5\right)
c. LaTeX: \left(x+9\right)\left(x+2\right)
d. LaTeX: \left(4x-1\right)\left(4x+1\right)
Once you have completed those, check your work here.
Let's look at another few examples:
4. LaTeX: \left(x+5\right)^2
while this one doesn't look like the ones we did on 3, it has the same principle. Be careful! A lot of students get this wrong because they think it's easy: Just square both terms: LaTeX: x^2+25. WRONG!!!!! DON'T DO THAT. Why is it not that way?
Remember: Raising something to the second power, means to times it by itself. But we need to realize: we need to times x+5 by itself. So, whenever you see something like this, we need to think: LaTeX: \left(x+5\right)^2=\:\left(x+5\right)\left(x+5\right)and we can simplify from there. When we distribute, we get: LaTeX: x^2+5x+5x+25.
Then we can combine the like terms and get: LaTeX: x^2+10x+25
Let's practice a few of these.
5.
a. LaTeX: \left(x-3\right)^2
b. LaTeX: \left(x-5\right)^2
c. LaTeX: \left(x-1\right)^2
d. LaTeX: \left(x+8\right)^2
Once you have simplified those, check your answers here.
We can now take this practice one step further.
6. Simplify the following expression: LaTeX: \left(x+5\right)^2-4
For this expression, we will simplify the first part - like we did above - and that will give us: LaTeX: x^2+10x+25\:-4. Then combine the numbers to give us: LaTeX: x^2+10x+21
This last question illustrates the two forms of a quadratic equation that we can have. When graphing quadratic equations, we can graph an equation like: LaTeX: f\left(x\right)=\left(x+3\right)^2-5or an equation that gives the same graph: LaTeX: f\left(x\right)=x^2+6x+4.
Our task today is to learn how to graph the first type: LaTeX: f\left(x\right)=\left(x+3\right)^2-5. We call this type vertex form of a quadratic. Let's first figure out what the parent function of this graph is.
Parent function: LaTeX: f\left(x\right)=x^2
Now create a table of values and plot those points on a graph:
x f(x)
-3
-2
-1
0
1
2
3
Your table of values and what the graph should look like is here.
We call this shape a parabola. That's a good word to remember. Write it down.
What is the rate of change between each value? Find the slope between each point using: LaTeX: \frac{y_2-y_1}{x_2-x_1}
It's a new rate of change we haven't discussed before. Our rates are: -5, -3, -1, 1, 3, 5. In fact, if we expand our graph, what do you think the next few rates are?
We can understand what the rates are by finding the difference in all our perfect squares. What are the first 8 perfect squares? LaTeX: 0^2=0,\:1^2=1,\:2^2=4,\:3^2=9and so forth.
Just looking at the results, we have: 0, 1, 4, 9, 16, 25, 36, 49, 64... What is the rate of change between these values? 1, 3, 5, 7, 9, 11, etc. Let's keep these numbers in mind whenever we are graphing quadratic functions.
Let's do another table of values and graph them.
LaTeX: g\left(x\right)=2x^2
x g(x)
-3
-2
-1
0
1
2
3
Plot these points on a graph. How is this graph related to the parent function: LaTeX: f\left(x\right)=x^2
The table and graph for this function is found here.
What is the rate of change between these values? How are those related to the rates of change from the parent function?
This graph has a rate of change of 2, 6, 10 on the right side of the vertex (the point at the bottom of the graph). If we refer back to the rate of change between our squares, how are these related to 2, 6, 10? We multiply the 1, 3, 5, 7, 9.... by the leading coefficient (the 2 in front of the equation) and that gives us: 2, 6, 10, 14, 18. The number in front determines how much you'll multiply the numbers 1, 3, 5, 7, 9... by.
So the graph of LaTeX: y=3x^2has rates of.... 3, 9, 15, 21.
What would the rates of LaTeX: y=\frac{1}{2}x^2be?
LaTeX: \frac{1}{2},\:\frac{3}{2},\:\frac{5}{2},\:\frac{7}{2}
So if we look at the graph of LaTeX: y=\frac{1}{2}x^2, the vertex is at (0, 0) and every unit we move to the right, our y-values increase by LaTeX: \frac{1}{2},\:\frac{3}{2},\:\frac{5}{2},\:\frac{7}{2}.
25.png
If the leading coefficient is a negative, let's say LaTeX: y=-3x^2, the graph would be decreasing by -3, -9, -15 etc.
26.png
So the only thing left we have to figure out is if we need to graph an equation like:
LaTeX: h\left(x\right)=-2\left(x-3\right)^2+5
From looking at this graph, the scale is going to be -2, -6, -10, but where is the vertex? What are the transformations in this graph? How is the parent function transformed?
Right 3, up 5. So the vertex of this parabola is at (3, 5) and the next points are going to be down 2, over 1. Then down 6, over 1. Then down 10, over 1.
27.png
Let's look at one more:
LaTeX: j\left(x\right)=3\left(x-5\right)^2-8
The vertex is where? (5, -8) (Remember, if it's with the x, think opposite)
28.png
If you have more questions, or need another explanation, I recorded this video last year that might help you out a little.
If you have questions, feel free to post them on the discussion board and I'll try to answer them as quickly as I can. (or if you know the answers to any questions on the discussion board, you can answer them and help your classmates).