Write a formula for quadratic function if its graph has the vertex at point (−2,5) and passes through the point (3,−10).
from the vertex,
y = a(x+2)^2 + 5
Now plug in your point to find a
To find the formula for a quadratic function with a vertex at (-2, 5) and passing through the point (3, -10), we need to use the vertex form of a quadratic function:
y = a(x - h)^2 + k
where h and k are the x-coordinate and y-coordinate of the vertex, respectively.
Using the information given, we have the vertex as (-2, 5), so h = -2 and k = 5.
Substituting these values into the equation, we get:
y = a(x - (-2))^2 + 5
Simplifying further, we have:
y = a(x + 2)^2 + 5
Now, we need to use the point (3, -10) to find the value of a.
Substituting x = 3 and y = -10 into the equation, we get:
-10 = a(3 + 2)^2 + 5
-10 = a(5)^2 + 5
-10 = 25a + 5
Subtracting 5 from both sides:
-15 = 25a
Dividing both sides by 25:
a = -15/25
Simplifying further:
a = -3/5
Therefore, the formula for the quadratic function with the given specifications is:
y = (-3/5)(x + 2)^2 + 5
To find a formula for a quadratic function, we need to consider the vertex form equation of a quadratic function, which is given by:
f(x) = a(x - h)^2 + k
Where:
- f(x) represents the quadratic function itself.
- a represents the coefficient that determines the shape and direction of the parabola.
- (h, k) represents the coordinates of the vertex.
Given that the vertex is at (-2, 5), we can substitute these values into the equation:
f(x) = a(x - (-2))^2 + 5
Simplifying this expression, we get:
f(x) = a(x + 2)^2 + 5
Now, to determine the value of a, we can use the point (3, -10) that the quadratic function passes through. By substituting this point into our equation, we get:
-10 = a(3 + 2)^2 + 5
Simplifying further:
-10 = a(5)^2 + 5
-10 = 25a + 5
-10 - 5 = 25a
-15 = 25a
Now, we can solve for a by dividing both sides by 25:
a = -15/25
a = -3/5
Substituting this value of a back into the equation, we get:
f(x) = (-3/5)(x + 2)^2 + 5
Therefore, the formula for the quadratic function whose graph has the vertex at (-2, 5) and passes through the point (3, -10) is:
f(x) = (-3/5)(x + 2)^2 + 5