It is given a quadratic function y=a(x-h)62+k.If its graph is a parabola with x=2 as its axis of symmetry,and it passes through the two points(3,7)and(4,11)
(a)what isthe value of the constant h?
(b)find the values of the constants a and k,and write down the quadratic function.
If its axis of symmetry is x=2, then h=2
7 = a + k
11 = 4a + k
3a = 4
a = 4/3
k = 17/3
y = 4/3(x-2)^2 + 17/3
To answer this question, we will use the given information about the quadratic function and the points it passes through.
(a) To find the value of the constant h, we know that the axis of symmetry of the parabola is x = 2. Since the axis of symmetry is given by the equation x = h, we can equate h with the given value of the axis of symmetry, which is h = 2.
(b) To find the values of the constants a and k and write down the quadratic function, we can use the given points that lie on the graph of the quadratic function.
Let's substitute the coordinates of the first point (3,7) into the equation of the quadratic function:
7 = a(3 - 2)^2 + k
Simplifying further:
7 = a(1)^2 + k
7 = a + k ------(Equation 1)
Now, let's substitute the coordinates of the second point (4,11) into the equation of the quadratic function:
11 = a(4 - 2)^2 + k
Simplifying further:
11 = a(2)^2 + k
11 = 4a + k ------(Equation 2)
We now have a system of two equations (Equation 1 and Equation 2) with two unknowns (a and k). We can solve this system of equations simultaneously to find the values of a and k.
Subtracting Equation 1 from Equation 2, we get:
11 - 7 = 4a + k - (a + k)
4 = 3a
Dividing both sides by 3:
a = 4/3
Substituting the value of a into Equation 1:
7 = (4/3) + k
Multiplying both sides by 3 to eliminate the fraction:
21 = 4 + 3k
Subtracting 4 from both sides:
17 = 3k
Dividing both sides by 3:
k = 17/3
Therefore, the values of the constants are:
a = 4/3
k = 17/3
Now that we have the values of a, h, and k, we can write down the quadratic function:
y = a(x - h)^2 + k
Substituting the values we found:
y = (4/3)(x - 2)^2 + 17/3
Hence, the quadratic function is y = (4/3)(x - 2)^2 + 17/3.