Math / Algebra

QuestionThe graph of linear function $f$ passes through the point $(1,-9)$ and has a slope of -3 .
What is the zero of $f$ ?
F 2 G 4 H -6 J -2

Studdy Solution

STEP 1

What is this asking? We're looking for the x-value where our line hits the x-axis, given a point it goes through and its slope! Watch out! Don't mix up the x and y values, and make sure you use the slope correctly!

STEP 2

1. Find the equation
2. Find the zero

STEP 3

Alright, let's start with the point-slope form of a linear equation, which is $y - y_1 = m(x - x_1)$ .
Remember, $(x_1, y_1)$ is our given point, and $m$ is our slope.
This formula is super handy because it tells us how much $y$ changes based on how much $x$ changes, all relative to a starting point!

STEP 4

We know our point is $(1, -9)$ , so $x_1 = 1$ and $y_1 = -9$ .
Our slope, $m$ , is $-3$ .
Let's plug these values into our point-slope form: $y - (-9) = -3(x - 1)$ .

STEP 5

Now, let's simplify! $y + 9 = -3x + 3$ .
See how we distributed the $-3$ ?
We multiplied it by both $x$ and $-1$ .

STEP 6

Let's isolate $y$ to get our equation in slope-intercept form ( $y = mx + b$ ).
We can do this by subtracting $9$ from both sides of the equation: $y = -3x + 3 - 9$ , which simplifies to $y = -3x - 6$ .
Awesome!

STEP 7

The "zero" of a function is just a fancy way of saying where the graph crosses the x-axis.
This happens when $y = 0$ .
So, let's substitute $0$ for $y$ in our equation: $0 = -3x - 6$ .

STEP 8

Now, we want to solve for $x$ .
Let's add $6$ to both sides of the equation to isolate the term with $x$ : $0 + 6 = -3x - 6 + 6$ , which simplifies to $6 = -3x$ .

STEP 9

Finally, let's divide both sides by $-3$ to get $x$ all by itself: $\frac{6}{-3} = \frac{-3x}{-3}$ .
This gives us $x = -2$ .
Boom! We found our zero!

STEP 10

The zero of $f$ is $-2$ , which corresponds to answer choice J.

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QuestionThe graph of linear function $f$ passes through the point $(1,-9)$ and has a slope of -3 .
What is the zero of $f$ ?
F 2 G 4 H -6 J -2

Studdy Solution

STEP 1

What is this asking? We're looking for the x-value where our line hits the x-axis, given a point it goes through and its slope! Watch out! Don't mix up the x and y values, and make sure you use the slope correctly!

STEP 2

1. Find the equation
2. Find the zero

STEP 3

STEP 4

We know our point is $(1, -9)$ , so $x_1 = 1$ and $y_1 = -9$ .
Our slope, $m$ , is $-3$ .
Let's plug these values into our point-slope form: $y - (-9) = -3(x - 1)$ .

STEP 5

Now, let's simplify! $y + 9 = -3x + 3$ .
See how we distributed the $-3$ ?
We multiplied it by both $x$ and $-1$ .

STEP 6

Let's isolate $y$ to get our equation in slope-intercept form ( $y = mx + b$ ).
We can do this by subtracting $9$ from both sides of the equation: $y = -3x + 3 - 9$ , which simplifies to $y = -3x - 6$ .
Awesome!

STEP 7

The "zero" of a function is just a fancy way of saying where the graph crosses the x-axis.
This happens when $y = 0$ .
So, let's substitute $0$ for $y$ in our equation: $0 = -3x - 6$ .

STEP 8

Now, we want to solve for $x$ .
Let's add $6$ to both sides of the equation to isolate the term with $x$ : $0 + 6 = -3x - 6 + 6$ , which simplifies to $6 = -3x$ .

STEP 9

Finally, let's divide both sides by $-3$ to get $x$ all by itself: $\frac{6}{-3} = \frac{-3x}{-3}$ .
This gives us $x = -2$ .
Boom! We found our zero!

STEP 10

The zero of $f$ is $-2$ , which corresponds to answer choice J.

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QuestionThe graph of linear function fff passes through the point (1,−9)(1,-9)(1,−9) and has a slope of -3 .What is the zero of fff ?F 2 G 4 H -6 J -2

Studdy Solution

STEP 1

What is this asking? We're looking for the *x*-value where our line hits the *x*-axis, given a point it goes through and its slope! Watch out! Don't mix up the *x* and *y* values, and make sure you use the slope correctly!

STEP 2

1. Find the equation2. Find the zero

STEP 3

STEP 4

We know our **point** is (1,−9)(1, -9)(1,−9), so x1=1x_1 = 1x1​=1 and y1=−9y_1 = -9y1​=−9.Our **slope**, mmm, is −3-3−3.Let's **plug** these values into our **point-slope form**: y−(−9)=−3(x−1)y - (-9) = -3(x - 1)y−(−9)=−3(x−1).

STEP 5

Now, let's **simplify**! y+9=−3x+3y + 9 = -3x + 3y+9=−3x+3.See how we distributed the −3-3−3?We multiplied it by both xxx and −1-1−1.

STEP 6

Let's **isolate** yyy to get our equation in **slope-intercept form** (y=mx+by = mx + by=mx+b).We can do this by **subtracting** 999 from both sides of the equation: y=−3x+3−9y = -3x + 3 - 9y=−3x+3−9, which simplifies to y=−3x−6y = -3x - 6y=−3x−6.Awesome!

STEP 7

The "**zero**" of a function is just a fancy way of saying where the graph crosses the *x*-axis.This happens when y=0y = 0y=0.So, let's **substitute** 000 for yyy in our equation: 0=−3x−60 = -3x - 60=−3x−6.

STEP 8

Now, we want to **solve for** xxx.Let's **add** 666 to both sides of the equation to **isolate** the term with xxx: 0+6=−3x−6+60 + 6 = -3x - 6 + 60+6=−3x−6+6, which simplifies to 6=−3x6 = -3x6=−3x.

STEP 9

Finally, let's **divide** both sides by −3-3−3 to get xxx all by itself: 6−3=−3x−3\frac{6}{-3} = \frac{-3x}{-3}−36​=−3−3x​.This gives us x=−2x = -2x=−2.Boom! We found our zero!

STEP 10

The zero of fff is −2-2−2, which corresponds to answer choice J.

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

QuestionThe graph of linear function $f$ passes through the point $(1,-9)$ and has a slope of -3 .
What is the zero of $f$ ?
F 2 G 4 H -6 J -2

What is this asking? We're looking for the x-value where our line hits the x-axis, given a point it goes through and its slope! Watch out! Don't mix up the x and y values, and make sure you use the slope correctly!

1. Find the equation
2. Find the zero

We know our point is $(1, -9)$ , so $x_1 = 1$ and $y_1 = -9$ .
Our slope, $m$ , is $-3$ .
Let's plug these values into our point-slope form: $y - (-9) = -3(x - 1)$ .

Now, let's simplify! $y + 9 = -3x + 3$ .
See how we distributed the $-3$ ?
We multiplied it by both $x$ and $-1$ .

Let's isolate $y$ to get our equation in slope-intercept form ( $y = mx + b$ ).
We can do this by subtracting $9$ from both sides of the equation: $y = -3x + 3 - 9$ , which simplifies to $y = -3x - 6$ .
Awesome!

The "zero" of a function is just a fancy way of saying where the graph crosses the x-axis.
This happens when $y = 0$ .
So, let's substitute $0$ for $y$ in our equation: $0 = -3x - 6$ .

Now, we want to solve for $x$ .
Let's add $6$ to both sides of the equation to isolate the term with $x$ : $0 + 6 = -3x - 6 + 6$ , which simplifies to $6 = -3x$ .

Finally, let's divide both sides by $-3$ to get $x$ all by itself: $\frac{6}{-3} = \frac{-3x}{-3}$ .
This gives us $x = -2$ .
Boom! We found our zero!

The zero of $f$ is $-2$ , which corresponds to answer choice J.