Math / Algebra

QuestionFor the relation: $y=2 x-4$ , sketch the graph of the relation and its inverse, then write the equation for the inverse.
Clear All Draw: $\square$
Inverse: $y=$ $\square$

Studdy Solution

STEP 1

What is this asking? We're taking a line, drawing it, then flipping it over the diagonal line $y=x$ to get its inverse, and then writing the equation of that flipped line! Watch out! Don't flip-flop the x and y without also solving for the new y!

STEP 2

1. Graph the original relation
2. Find the inverse relation
3. Graph the inverse relation

STEP 3

Alright, let's graph this bad boy!
Our equation is $y = 2x - 4$ .
This is in slope-intercept form, which is $y = mx + b$ , where m is the slope and b is the y-intercept.

STEP 4

Here, our slope is $m = \mathbf{2}$ and our y-intercept is $b = \mathbf{-4}$ .
The y-intercept tells us that the line crosses the y-axis at the point $(0, -4)$ .
Plot that point!

STEP 5

The slope is $2$ , which we can write as $\frac{\mathbf{2}}{\mathbf{1}}$ .
Remember, slope is rise over run.
So, from our y-intercept $(0, -4)$ , we go up 2 and over 1 to the right.
This gives us another point on the line: $(1, -2)$ .
Plot that point!
Now draw a straight line through the two points.
Boom! We've graphed our original line.

STEP 6

To find the inverse, we swap x and y in the original equation.
So, $y = 2x - 4$ becomes $x = 2y - 4$ .

STEP 7

Now, we need to solve for y.
First, let's add 4 to both sides of the equation: $x + 4 = 2y - 4 + 4$ , which simplifies to $x + 4 = 2y$ .

STEP 8

Next, we divide both sides by $2$ to isolate y: $\frac{x + 4}{2} = \frac{2y}{2}$ .
This simplifies to $y = \frac{x + 4}{2}$ , or $y = \frac{1}{2}x + 2$ .
This is the equation of our inverse relation!

STEP 9

Now, let's graph the inverse.
Our equation is $y = \frac{1}{2}x + 2$ .
This is also in slope-intercept form, where our slope is $m = \frac{\mathbf{1}}{\mathbf{2}}$ and our y-intercept is $b = \mathbf{2}$ .

STEP 10

The y-intercept tells us the line crosses the y-axis at $(0, 2)$ .
Plot that point!

STEP 11

Our slope is $\frac{1}{2}$ , so from the y-intercept, we go up 1 and over 2 to the right.
This gives us the point $(2, 3)$ .
Plot that point!
Draw a straight line through these two points, and there you have it – the graph of the inverse!

STEP 12

The graph of $y = 2x - 4$ is a line passing through $(0, -4)$ and $(1, -2)$ .
The inverse relation is $y = \frac{1}{2}x + 2$ , and its graph is a line passing through $(0, 2)$ and $(2, 3)$ .

Was this helpful?

QuestionFor the relation: $y=2 x-4$ , sketch the graph of the relation and its inverse, then write the equation for the inverse.
Clear All Draw: $\square$
Inverse: $y=$ $\square$

Studdy Solution

STEP 1

What is this asking? We're taking a line, drawing it, then flipping it over the diagonal line $y=x$ to get its inverse, and then writing the equation of that flipped line! Watch out! Don't flip-flop the x and y without also solving for the new y!

STEP 2

1. Graph the original relation
2. Find the inverse relation
3. Graph the inverse relation

STEP 3

Alright, let's graph this bad boy!
Our equation is $y = 2x - 4$ .
This is in slope-intercept form, which is $y = mx + b$ , where m is the slope and b is the y-intercept.

STEP 4

Here, our slope is $m = \mathbf{2}$ and our y-intercept is $b = \mathbf{-4}$ .
The y-intercept tells us that the line crosses the y-axis at the point $(0, -4)$ .
Plot that point!

STEP 5

STEP 6

To find the inverse, we swap x and y in the original equation.
So, $y = 2x - 4$ becomes $x = 2y - 4$ .

STEP 7

Now, we need to solve for y.
First, let's add 4 to both sides of the equation: $x + 4 = 2y - 4 + 4$ , which simplifies to $x + 4 = 2y$ .

STEP 8

Next, we divide both sides by $2$ to isolate y: $\frac{x + 4}{2} = \frac{2y}{2}$ .
This simplifies to $y = \frac{x + 4}{2}$ , or $y = \frac{1}{2}x + 2$ .
This is the equation of our inverse relation!

STEP 9

Now, let's graph the inverse.
Our equation is $y = \frac{1}{2}x + 2$ .
This is also in slope-intercept form, where our slope is $m = \frac{\mathbf{1}}{\mathbf{2}}$ and our y-intercept is $b = \mathbf{2}$ .

STEP 10

The y-intercept tells us the line crosses the y-axis at $(0, 2)$ .
Plot that point!

STEP 11

Our slope is $\frac{1}{2}$ , so from the y-intercept, we go up 1 and over 2 to the right.
This gives us the point $(2, 3)$ .
Plot that point!
Draw a straight line through these two points, and there you have it – the graph of the inverse!

STEP 12

The graph of $y = 2x - 4$ is a line passing through $(0, -4)$ and $(1, -2)$ .
The inverse relation is $y = \frac{1}{2}x + 2$ , and its graph is a line passing through $(0, 2)$ and $(2, 3)$ .

Get Studdy

Studdy solves anything!

Start learning now

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

QuestionFor the relation: y=2x−4y=2 x-4y=2x−4, sketch the graph of the relation and its inverse, then write the equation for the inverse.Clear All Draw: □\square□Inverse: y=y=y= □\square□

Studdy Solution

STEP 1

What is this asking? We're taking a line, drawing it, then flipping it over the diagonal line y=xy=xy=x to get its inverse, and then writing the equation of that flipped line! Watch out! Don't flip-flop the *x* and *y* without also solving for the new *y*!

STEP 2

1. Graph the original relation2. Find the inverse relation3. Graph the inverse relation

STEP 3

Alright, let's **graph** this bad boy!Our equation is y=2x−4y = 2x - 4y=2x−4.This is in **slope-intercept form**, which is y=mx+by = mx + by=mx+b, where *m* is the **slope** and *b* is the **y-intercept**.

STEP 4

Here, our **slope** is m=2m = \mathbf{2}m=2 and our **y-intercept** is b=−4b = \mathbf{-4}b=−4.The y-intercept tells us that the line crosses the y-axis at the point (0,−4)(0, -4)(0,−4).Plot that point!

STEP 5

STEP 6

To find the inverse, we **swap** *x* and *y* in the original equation.So, y=2x−4y = 2x - 4y=2x−4 becomes x=2y−4x = 2y - 4x=2y−4.

STEP 7

Now, we need to **solve for** *y*.First, let's **add 4** to both sides of the equation: x+4=2y−4+4x + 4 = 2y - 4 + 4x+4=2y−4+4, which simplifies to x+4=2yx + 4 = 2yx+4=2y.

STEP 8

Next, we **divide both sides** by 222 to isolate *y*: x+42=2y2\frac{x + 4}{2} = \frac{2y}{2}2x+4​=22y​.This simplifies to y=x+42y = \frac{x + 4}{2}y=2x+4​, or y=12x+2y = \frac{1}{2}x + 2y=21​x+2.This is the equation of our inverse relation!

STEP 9

Now, let's **graph** the inverse.Our equation is y=12x+2y = \frac{1}{2}x + 2y=21​x+2.This is also in **slope-intercept form**, where our **slope** is m=12m = \frac{\mathbf{1}}{\mathbf{2}}m=21​ and our **y-intercept** is b=2b = \mathbf{2}b=2.

STEP 10

The **y-intercept** tells us the line crosses the y-axis at (0,2)(0, 2)(0,2).Plot that point!

STEP 11

Our **slope** is 12\frac{1}{2}21​, so from the **y-intercept**, we go **up 1** and **over 2** to the **right**.This gives us the point (2,3)(2, 3)(2,3).Plot that point!Draw a straight line through these two points, and there you have it – the graph of the inverse!

STEP 12

The graph of y=2x−4y = 2x - 4y=2x−4 is a line passing through (0,−4)(0, -4)(0,−4) and (1,−2)(1, -2)(1,−2).The inverse relation is y=12x+2y = \frac{1}{2}x + 2y=21​x+2, and its graph is a line passing through (0,2)(0, 2)(0,2) and (2,3)(2, 3)(2,3).

Get Studdy

Studdy solves anything!

Start learning now

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

QuestionFor the relation: $y=2 x-4$ , sketch the graph of the relation and its inverse, then write the equation for the inverse.
Clear All Draw: $\square$
Inverse: $y=$ $\square$

What is this asking? We're taking a line, drawing it, then flipping it over the diagonal line $y=x$ to get its inverse, and then writing the equation of that flipped line! Watch out! Don't flip-flop the x and y without also solving for the new y!

1. Graph the original relation
2. Find the inverse relation
3. Graph the inverse relation

Alright, let's graph this bad boy!
Our equation is $y = 2x - 4$ .
This is in slope-intercept form, which is $y = mx + b$ , where m is the slope and b is the y-intercept.

Here, our slope is $m = \mathbf{2}$ and our y-intercept is $b = \mathbf{-4}$ .
The y-intercept tells us that the line crosses the y-axis at the point $(0, -4)$ .
Plot that point!

To find the inverse, we swap x and y in the original equation.
So, $y = 2x - 4$ becomes $x = 2y - 4$ .

Now, we need to solve for y.
First, let's add 4 to both sides of the equation: $x + 4 = 2y - 4 + 4$ , which simplifies to $x + 4 = 2y$ .

Next, we divide both sides by $2$ to isolate y: $\frac{x + 4}{2} = \frac{2y}{2}$ .
This simplifies to $y = \frac{x + 4}{2}$ , or $y = \frac{1}{2}x + 2$ .
This is the equation of our inverse relation!

Now, let's graph the inverse.
Our equation is $y = \frac{1}{2}x + 2$ .
This is also in slope-intercept form, where our slope is $m = \frac{\mathbf{1}}{\mathbf{2}}$ and our y-intercept is $b = \mathbf{2}$ .

The y-intercept tells us the line crosses the y-axis at $(0, 2)$ .
Plot that point!

Our slope is $\frac{1}{2}$ , so from the y-intercept, we go up 1 and over 2 to the right.
This gives us the point $(2, 3)$ .
Plot that point!
Draw a straight line through these two points, and there you have it – the graph of the inverse!

The graph of $y = 2x - 4$ is a line passing through $(0, -4)$ and $(1, -2)$ .
The inverse relation is $y = \frac{1}{2}x + 2$ , and its graph is a line passing through $(0, 2)$ and $(2, 3)$ .