Math / Algebra

QuestionGraph the following function by starting with the graph of $y=x^{2}$ and using transformations (shifting, compressing, stretching, and/or reflection). $f(x)=\frac{2}{3} x^{2}$
Use the graphing tool to graph the function.

Studdy Solution

STEP 1

What is this asking? We need to graph $f(x) = \frac{2}{3}x^2$ by transforming the simpler graph of $y = x^2$ . Watch out! Don't mix up compressing and stretching!
Also, remember that changes inside the $x^2$ part move the graph horizontally, while changes outside move it vertically.

STEP 2

1. Set up the basic parabola.
2. Apply the transformation.

STEP 3

Alright, let's kick things off with our basic parabola, $y = x^2$ !
We're going to use this as our foundation and transform it into the desired graph.
Think of it like starting with a plain pizza dough and adding all the delicious toppings!

STEP 4

Remember what $y = x^2$ looks like?
It's that beautiful, symmetrical U-shape centered right at the origin $(0, 0)$ .
When $x = 1$ , $y = 1^2 = 1$ .
When $x = -1$ , $y = (-1)^2 = 1$ .
When $x = 2$ , $y = 2^2 = 4$ .
And so on!

STEP 5

Now, let's look at our target function: $f(x) = \frac{2}{3}x^2$ .
See that $\frac{2}{3}$ hanging out in front?
That's our transformation factor!
Since it's outside the $x^2$ , it's going to affect the graph vertically.

STEP 6

Since $\frac{2}{3}$ is between zero and one, this means we're going to compress our parabola vertically.
Imagine gently pushing down on the parabola, making it a bit wider.

STEP 7

Let's see how this compression works with an example.
If we take $x = 1$ , then $f(1) = \frac{2}{3}(1)^2 = \frac{2}{3}$ .
See how the $y$ -value is $\frac{2}{3}$ of what it was for the basic parabola ( $y = 1$ )?
The same thing happens for all other $x$ values!
If $x = 2$ , then $f(2) = \frac{2}{3}(2)^2 = \frac{2}{3} \cdot 4 = \frac{8}{3}$ .
Again, the $y$ -value is $\frac{2}{3}$ of what it would be for $y = x^2$ (which would be $4$ ).

STEP 8

So, to graph $f(x) = \frac{2}{3}x^2$ , take your basic parabola $y = x^2$ and compress it vertically by a factor of $\frac{2}{3}$ .
It'll still be centered at the origin $(0, 0)$ , but it'll be a bit wider than the basic parabola.

STEP 9

The graph of $f(x) = \frac{2}{3}x^2$ is a vertically compressed version of the graph of $y = x^2$ , with the compression factor being $\frac{2}{3}$ .
The graph still passes through the origin $(0, 0)$ , but other points are closer to the $x$ -axis than in the basic parabola.

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QuestionGraph the following function by starting with the graph of $y=x^{2}$ and using transformations (shifting, compressing, stretching, and/or reflection). $f(x)=\frac{2}{3} x^{2}$
Use the graphing tool to graph the function.

Studdy Solution

STEP 1

What is this asking? We need to graph $f(x) = \frac{2}{3}x^2$ by transforming the simpler graph of $y = x^2$ . Watch out! Don't mix up compressing and stretching!
Also, remember that changes inside the $x^2$ part move the graph horizontally, while changes outside move it vertically.

STEP 2

1. Set up the basic parabola.
2. Apply the transformation.

STEP 3

Alright, let's kick things off with our basic parabola, $y = x^2$ !
We're going to use this as our foundation and transform it into the desired graph.
Think of it like starting with a plain pizza dough and adding all the delicious toppings!

STEP 4

Remember what $y = x^2$ looks like?
It's that beautiful, symmetrical U-shape centered right at the origin $(0, 0)$ .
When $x = 1$ , $y = 1^2 = 1$ .
When $x = -1$ , $y = (-1)^2 = 1$ .
When $x = 2$ , $y = 2^2 = 4$ .
And so on!

STEP 5

Now, let's look at our target function: $f(x) = \frac{2}{3}x^2$ .
See that $\frac{2}{3}$ hanging out in front?
That's our transformation factor!
Since it's outside the $x^2$ , it's going to affect the graph vertically.

STEP 6

Since $\frac{2}{3}$ is between zero and one, this means we're going to compress our parabola vertically.
Imagine gently pushing down on the parabola, making it a bit wider.

STEP 7

STEP 8

So, to graph $f(x) = \frac{2}{3}x^2$ , take your basic parabola $y = x^2$ and compress it vertically by a factor of $\frac{2}{3}$ .
It'll still be centered at the origin $(0, 0)$ , but it'll be a bit wider than the basic parabola.

STEP 9

The graph of $f(x) = \frac{2}{3}x^2$ is a vertically compressed version of the graph of $y = x^2$ , with the compression factor being $\frac{2}{3}$ .
The graph still passes through the origin $(0, 0)$ , but other points are closer to the $x$ -axis than in the basic parabola.

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QuestionGraph the following function by starting with the graph of y=x2y=x^{2}y=x2 and using transformations (shifting, compressing, stretching, and/or reflection). f(x)=23x2f(x)=\frac{2}{3} x^{2}f(x)=32​x2Use the graphing tool to graph the function.

Studdy Solution

STEP 1

STEP 2

1. Set up the basic parabola.2. Apply the transformation.

STEP 3

Alright, let's kick things off with our **basic parabola**, y=x2y = x^2y=x2!We're going to use this as our **foundation** and **transform** it into the desired graph.Think of it like starting with a plain pizza dough and adding all the delicious toppings!

STEP 4

Remember what y=x2y = x^2y=x2 looks like?It's that **beautiful, symmetrical U-shape** centered right at the **origin** (0,0)(0, 0)(0,0).When x=1x = 1x=1, y=12=1y = 1^2 = 1y=12=1.When x=−1x = -1x=−1, y=(−1)2=1y = (-1)^2 = 1y=(−1)2=1.When x=2x = 2x=2, y=22=4y = 2^2 = 4y=22=4.And so on!

STEP 5

Now, let's look at our target function: f(x)=23x2f(x) = \frac{2}{3}x^2f(x)=32​x2.See that 23\frac{2}{3}32​ hanging out in front?That's our **transformation factor**!Since it's *outside* the x2x^2x2, it's going to affect the graph **vertically**.

STEP 6

Since 23\frac{2}{3}32​ is between **zero** and **one**, this means we're going to **compress** our parabola **vertically**.Imagine gently pushing down on the parabola, making it a bit wider.

STEP 7

STEP 8

So, to graph f(x)=23x2f(x) = \frac{2}{3}x^2f(x)=32​x2, take your basic parabola y=x2y = x^2y=x2 and **compress it vertically** by a factor of 23\frac{2}{3}32​.It'll still be centered at the origin (0,0)(0, 0)(0,0), but it'll be a bit wider than the basic parabola.

STEP 9

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionGraph the following function by starting with the graph of $y=x^{2}$ and using transformations (shifting, compressing, stretching, and/or reflection). $f(x)=\frac{2}{3} x^{2}$
Use the graphing tool to graph the function.

1. Set up the basic parabola.
2. Apply the transformation.

Alright, let's kick things off with our basic parabola, $y = x^2$ !
We're going to use this as our foundation and transform it into the desired graph.
Think of it like starting with a plain pizza dough and adding all the delicious toppings!

Remember what $y = x^2$ looks like?
It's that beautiful, symmetrical U-shape centered right at the origin $(0, 0)$ .
When $x = 1$ , $y = 1^2 = 1$ .
When $x = -1$ , $y = (-1)^2 = 1$ .
When $x = 2$ , $y = 2^2 = 4$ .
And so on!

Now, let's look at our target function: $f(x) = \frac{2}{3}x^2$ .
See that $\frac{2}{3}$ hanging out in front?
That's our transformation factor!
Since it's outside the $x^2$ , it's going to affect the graph vertically.

Since $\frac{2}{3}$ is between zero and one, this means we're going to compress our parabola vertically.
Imagine gently pushing down on the parabola, making it a bit wider.

So, to graph $f(x) = \frac{2}{3}x^2$ , take your basic parabola $y = x^2$ and compress it vertically by a factor of $\frac{2}{3}$ .
It'll still be centered at the origin $(0, 0)$ , but it'll be a bit wider than the basic parabola.