the graphs of the function with equation y= a(x+5)^2=q passes through the points (-6,2) and (-3,20).
Determine whether the function has a maximum or minimum value
I will assume you mean
y= a(x+5)^2 + q , ( + and = are on the same key)
for the point (-6,2) ,
2 = a(-1)^2 + q ---> a + q = 2
for the point (-3,20) ,
20 = a(4) + q ---> 4a + q = 20
solve these two equations, I suggest subtracting them
THis is a really nice question : )
If you sub one of the points into the equation (for x and y) you obtain equation 1 in values of q and a.
If you then sub the second point (-3, 20) into the original equation
20 = a(- 3 + 5)^2 + q you obtain equation 2.
When you use equations 1 and 2 and "subtract" them, you can find the value of "a". If "a" equals a positive number then the quadratic opens upwards (and is a minimum), if "a" equals a negative number then the quadratic opens downwards (and is a minimum) : )
To determine whether the function has a maximum or minimum value, we need to analyze the equation and the given points.
The given equation is y = a(x + 5)^2 + q.
Let's substitute the coordinates of the given points into the equation and try to solve for the variables a and q.
For the point (-6, 2):
2 = a(-6 + 5)^2 + q
2 = a(1)^2 + q
2 = a + q ----- (Equation 1)
For the point (-3, 20):
20 = a(-3 + 5)^2 + q
20 = a(2)^2 + q
20 = 4a + q ----- (Equation 2)
Now, we have a system of equations with two variables (a and q). Let's solve this system to find the values of a and q.
By subtracting Equation 1 from Equation 2, we can eliminate q:
20 - 2 = 4a + q - (a + q)
18 = 4a - a
18 = 3a
a = 6
By substituting the value of a into Equation 1, we can find q:
2 = 6 + q
q = -4
So, we have found the values a = 6 and q = -4.
Now, let's examine the parabola represented by the equation y = 6(x + 5)^2 - 4.
The coefficient (a) is positive, which indicates that the parabola opens upwards. This means that the function has a minimum value. The vertex of the parabola represents the minimum point of the function.
The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
In our case, h = -5 and k = -4. So, the vertex of the parabola is (-5, -4).
Therefore, the function y = 6(x + 5)^2 - 4 has a minimum value.