Write the formula for absolute value function if its graph has the vertex at point ( 1/3 ,−3) and passes through the point (1,1).
Sorry, that is incorrect
It is all right except that there is no squaring
y=a|x-1/3|-3
1=a|1-1/3|-3
4=2/3a
a=6
Anwer: y=6|x-1/3|-3
bro just tell us the answer
*answer
To find the formula for the absolute value function given the vertex and a point on the graph, you can follow these steps:
Step 1: Recognize that the absolute value function can be represented by two linear equations since it changes direction at the vertex.
Step 2: Determine the equation for the segment of the absolute value function to the left of the vertex using the point-slope form.
The equation for the left segment of the absolute value function is:
y = m₁x + b₁
where m₁ is the slope and b₁ is the y-intercept.
Step 3: To find the slope (m₁), we can use the formula:
m₁ = (y₂ - y₁) / (x₂ - x₁)
where (x₁, y₁) is the vertex (1/3, -3) and (x₂, y₂) is the given point (1, 1).
m₁ = (1 - (-3)) / (1 - 1/3)
= 4 / (3/3 - 1/3)
= 4 / (2/3)
= 6
Step 4: Substitute the slope value into the equation:
y = 6x + b₁
Step 5: Calculate the y-intercept (b₁) by substituting the coordinates of the given point (1, 1) into the equation:
1 = 6(1) + b₁
1 = 6 + b₁
b₁ = 1 - 6
b₁ = -5
Step 6: Substitute the y-intercept (b₁) value into the equation:
y = 6x - 5
Step 7: The equation for the segment to the right of the vertex is a reflection of the left segment, so it will have the same slope but a different y-intercept.
The equation for the right segment of the absolute value function is:
y = 6x + b₂
where b₂ is the y-intercept for the right segment.
Step 8: Since the vertex is the lowest point on the graph, the right segment opens upwards. Therefore, the y-intercept for the right segment will be the same as the left segment, but positive:
b₂ = -(-5) = 5
Step 9: Substitute the values into the equation:
y = 6x + 5
Step 10: Combine the equations for both segments:
| x | = {
6x - 5 if x ≤ 1/3,
6x + 5 if x > 1/3
}
So, the formula for the absolute value function with a vertex at (1/3, -3) and passing through (1, 1) is:
| x | = {
6x - 5 if x ≤ 1/3,
6x + 5 if x > 1/3
}
poop
y = a(x -1/3) - 3
Now use the point to find a
it would be "y=a(x-1/3)^2-3"
then 1=a(1-1/3)^2-3
then 4=4/9a
then a=9
therefore y=9|x-1/3|^2-3