Solved on Oct 24, 2023

Find the equation of an absolute value function with vertex at $(-3, -1)$ , opening upwards, and passing through $(0, 2)$ .

STEP 1

Assumptions1. The vertex of the absolute value graph is at (-3, -1) . The graph opens upwards3. The graph has shifted to the left3 units and down one unit4. The graph goes through the point (0,)

STEP 2

The general form of an absolute value function is $y = a|x-h| + k$ where (h, k) is the vertex of the graph, 'a' determines the direction and steepness of the graph.

STEP 3

Since the graph opens upwards, 'a' is positive. And since the graph has shifted to the left3 units and down one unit, h = -3 and k = -1. So, the function becomes $y = a|x+3| -1$

STEP 4

Now, we need to find the value of 'a'. We can do this by substituting the coordinates of the given point (0,2) into the function.
$2 = a|0+3| -1$

STEP 5

olve the equation for 'a'.
$2 = a*3 -1$

STEP 6

Rearrange the equation to isolate 'a' on one side.
$a = (2+1)/3$

STEP 7

Calculate the value of 'a'.
$a =3/3 =1$

STEP 8

Now that we have the value of 'a', we can write the equation of the function.
$y = |x+3| -1$ The equation of the function is $y = |x+3| -1$ .

Was this helpful?

Find the equation of an absolute value function with vertex at $(-3, -1)$ , opening upwards, and passing through $(0, 2)$ .

STEP 1

Assumptions1. The vertex of the absolute value graph is at (-3, -1) . The graph opens upwards3. The graph has shifted to the left3 units and down one unit4. The graph goes through the point (0,)

STEP 2

The general form of an absolute value function is $y = a|x-h| + k$ where (h, k) is the vertex of the graph, 'a' determines the direction and steepness of the graph.

STEP 3

Since the graph opens upwards, 'a' is positive. And since the graph has shifted to the left3 units and down one unit, h = -3 and k = -1. So, the function becomes $y = a|x+3| -1$

STEP 4

Now, we need to find the value of 'a'. We can do this by substituting the coordinates of the given point (0,2) into the function.
$2 = a|0+3| -1$

STEP 5

olve the equation for 'a'.
$2 = a*3 -1$

STEP 6

Rearrange the equation to isolate 'a' on one side.
$a = (2+1)/3$

STEP 7

Calculate the value of 'a'.
$a =3/3 =1$

STEP 8

Now that we have the value of 'a', we can write the equation of the function.
$y = |x+3| -1$ The equation of the function is $y = |x+3| -1$ .

Start learning now

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

Find the equation of an absolute value function with vertex at (−3,−1)(-3, -1)(−3,−1), opening upwards, and passing through (0,2)(0, 2)(0,2).

STEP 1

Assumptions1. The vertex of the absolute value graph is at (-3, -1) . The graph opens upwards3. The graph has shifted to the left3 units and down one unit4. The graph goes through the point (0,)

STEP 2

The general form of an absolute value function isy=a∣x−h∣+ky = a|x-h| + ky=a∣x−h∣+kwhere (h, k) is the vertex of the graph, 'a' determines the direction and steepness of the graph.

STEP 3

Since the graph opens upwards, 'a' is positive. And since the graph has shifted to the left3 units and down one unit, h = -3 and k = -1. So, the function becomesy=a∣x+3∣−1y = a|x+3| -1y=a∣x+3∣−1

STEP 4

Now, we need to find the value of 'a'. We can do this by substituting the coordinates of the given point (0,2) into the function.2=a∣0+3∣−12 = a|0+3| -12=a∣0+3∣−1

STEP 5

olve the equation for 'a'.2=a∗3−12 = a*3 -12=a∗3−1

STEP 6

Rearrange the equation to isolate 'a' on one side.a=(2+1)/3a = (2+1)/3a=(2+1)/3

STEP 7

Calculate the value of 'a'.a=3/3=1a =3/3 =1a=3/3=1

STEP 8

Now that we have the value of 'a', we can write the equation of the function.y=∣x+3∣−1y = |x+3| -1y=∣x+3∣−1The equation of the function is y=∣x+3∣−1y = |x+3| -1y=∣x+3∣−1.

Start learning now

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

Find the equation of an absolute value function with vertex at $(-3, -1)$ , opening upwards, and passing through $(0, 2)$ .

The general form of an absolute value function is $y = a|x-h| + k$ where (h, k) is the vertex of the graph, 'a' determines the direction and steepness of the graph.

Since the graph opens upwards, 'a' is positive. And since the graph has shifted to the left3 units and down one unit, h = -3 and k = -1. So, the function becomes $y = a|x+3| -1$

Now, we need to find the value of 'a'. We can do this by substituting the coordinates of the given point (0,2) into the function.
$2 = a|0+3| -1$

olve the equation for 'a'.
$2 = a*3 -1$

Rearrange the equation to isolate 'a' on one side.
$a = (2+1)/3$

Calculate the value of 'a'.
$a =3/3 =1$

Now that we have the value of 'a', we can write the equation of the function.
$y = |x+3| -1$ The equation of the function is $y = |x+3| -1$ .