Math / Algebra

QuestionDescribe how the graph of $y=2 x$ is the same and different from the graph of $y=2 x-7$ Explain or show your reasoning.

Studdy Solution

STEP 1

What is this asking? How are the graphs of $y = 2x$ and $y = 2x - 7$ similar and different? Watch out! Don't forget to discuss both similarities and differences!

STEP 2

1. Analyze the slope
2. Analyze the y-intercept
3. Summarize similarities and differences

STEP 3

Let's investigate the slope of $y = 2x$ .
Remember, the general form of a linear equation is $y = mx + b$ , where $m$ is the slope.
In our case, $m = 2$ .
This tells us that for every 1 unit increase in $x$ , $y$ increases by 2 units.
It's a positive slope, so the line goes up as we move from left to right!

STEP 4

Now, let's peek at the slope of $y = 2x - 7$ .
Here, $m$ is also 2!
So, for every 1 unit increase in $x$ , $y$ also increases by 2 units.

STEP 5

The y-intercept is the value of $y$ when $x = 0$ .
For $y = 2x$ , if we plug in $x = 0$ , we get $y = 2 \cdot 0 = 0$ .
So, the y-intercept is 0, meaning the line crosses the y-axis at the origin $(0, 0)$ .

STEP 6

For $y = 2x - 7$ , if we plug in $x = 0$ , we get $y = 2 \cdot 0 - 7 = -7$ .
So, the y-intercept is -7.
This line crosses the y-axis at the point $(0, -7)$ .

STEP 7

Similarities: Both lines have the same slope of $2$ , meaning they have the same steepness and are parallel to each other.

STEP 8

Differences: The lines have different y-intercepts.
The first line crosses the y-axis at $y = 0$ , while the second line crosses at $y = -7$ .
This means the second line is shifted 7 units down compared to the first line.

STEP 9

Both lines have the same slope of $2$ making them parallel, but they have different y-intercepts: $0$ for $y=2x$ and $-7$ for $y=2x-7$ .
This means $y=2x-7$ is shifted $7$ units down from $y=2x$ .

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QuestionDescribe how the graph of $y=2 x$ is the same and different from the graph of $y=2 x-7$ Explain or show your reasoning.

Studdy Solution

STEP 1

What is this asking? How are the graphs of $y = 2x$ and $y = 2x - 7$ similar and different? Watch out! Don't forget to discuss both similarities and differences!

STEP 2

1. Analyze the slope
2. Analyze the y-intercept
3. Summarize similarities and differences

STEP 3

STEP 4

Now, let's peek at the slope of $y = 2x - 7$ .
Here, $m$ is also 2!
So, for every 1 unit increase in $x$ , $y$ also increases by 2 units.

STEP 5

The y-intercept is the value of $y$ when $x = 0$ .
For $y = 2x$ , if we plug in $x = 0$ , we get $y = 2 \cdot 0 = 0$ .
So, the y-intercept is 0, meaning the line crosses the y-axis at the origin $(0, 0)$ .

STEP 6

For $y = 2x - 7$ , if we plug in $x = 0$ , we get $y = 2 \cdot 0 - 7 = -7$ .
So, the y-intercept is -7.
This line crosses the y-axis at the point $(0, -7)$ .

STEP 7

Similarities: Both lines have the same slope of $2$ , meaning they have the same steepness and are parallel to each other.

STEP 8

Differences: The lines have different y-intercepts.
The first line crosses the y-axis at $y = 0$ , while the second line crosses at $y = -7$ .
This means the second line is shifted 7 units down compared to the first line.

STEP 9

Both lines have the same slope of $2$ making them parallel, but they have different y-intercepts: $0$ for $y=2x$ and $-7$ for $y=2x-7$ .
This means $y=2x-7$ is shifted $7$ units down from $y=2x$ .

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QuestionDescribe how the graph of y=2xy=2 xy=2x is the same and different from the graph of y=2x−7y=2 x-7y=2x−7 Explain or show your reasoning.

Studdy Solution

STEP 1

What is this asking? How are the graphs of y=2xy = 2xy=2x and y=2x−7y = 2x - 7y=2x−7 similar and different? Watch out! Don't forget to discuss both similarities *and* differences!

STEP 2

1. Analyze the slope2. Analyze the y-intercept3. Summarize similarities and differences

STEP 3

STEP 4

Now, let's **peek** at the **slope** of y=2x−7y = 2x - 7y=2x−7.Here, mmm is *also* **2**!So, for every **1 unit increase** in xxx, yyy *also* increases by **2 units**.

STEP 5

The **y-intercept** is the value of yyy when x=0x = 0x=0.For y=2xy = 2xy=2x, if we plug in x=0x = 0x=0, we get y=2⋅0=0y = 2 \cdot 0 = 0y=2⋅0=0.So, the **y-intercept** is **0**, meaning the line crosses the y-axis at the **origin** (0,0)(0, 0)(0,0).

STEP 6

For y=2x−7y = 2x - 7y=2x−7, if we plug in x=0x = 0x=0, we get y=2⋅0−7=−7y = 2 \cdot 0 - 7 = -7y=2⋅0−7=−7.So, the **y-intercept** is **-7**.This line crosses the y-axis at the point (0,−7)(0, -7)(0,−7).

STEP 7

**Similarities**: Both lines have the *same* **slope** of 222, meaning they have the same **steepness** and are **parallel** to each other.

STEP 8

**Differences**: The lines have *different* **y-intercepts**.The first line crosses the y-axis at y=0y = 0y=0, while the second line crosses at y=−7y = -7y=−7.This means the second line is shifted **7 units down** compared to the first line.

STEP 9

Both lines have the same slope of 222 making them parallel, but they have different y-intercepts: 000 for y=2xy=2xy=2x and −7-7−7 for y=2x−7y=2x-7y=2x−7.This means y=2x−7y=2x-7y=2x−7 is shifted 777 units down from y=2xy=2xy=2x.

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

QuestionDescribe how the graph of $y=2 x$ is the same and different from the graph of $y=2 x-7$ Explain or show your reasoning.

What is this asking? How are the graphs of $y = 2x$ and $y = 2x - 7$ similar and different? Watch out! Don't forget to discuss both similarities and differences!

1. Analyze the slope
2. Analyze the y-intercept
3. Summarize similarities and differences

Now, let's peek at the slope of $y = 2x - 7$ .
Here, $m$ is also 2!
So, for every 1 unit increase in $x$ , $y$ also increases by 2 units.

The y-intercept is the value of $y$ when $x = 0$ .
For $y = 2x$ , if we plug in $x = 0$ , we get $y = 2 \cdot 0 = 0$ .
So, the y-intercept is 0, meaning the line crosses the y-axis at the origin $(0, 0)$ .

For $y = 2x - 7$ , if we plug in $x = 0$ , we get $y = 2 \cdot 0 - 7 = -7$ .
So, the y-intercept is -7.
This line crosses the y-axis at the point $(0, -7)$ .

Similarities: Both lines have the same slope of $2$ , meaning they have the same steepness and are parallel to each other.

Differences: The lines have different y-intercepts.
The first line crosses the y-axis at $y = 0$ , while the second line crosses at $y = -7$ .
This means the second line is shifted 7 units down compared to the first line.

Both lines have the same slope of $2$ making them parallel, but they have different y-intercepts: $0$ for $y=2x$ and $-7$ for $y=2x-7$ .
This means $y=2x-7$ is shifted $7$ units down from $y=2x$ .