Axis symmetry x=3 passing through (3,1) and (1,9)
Not sure how to solve
(3,1) is the vertex of the parabola.
(1,9) and (5,9)
y = ax^2 + bx + c
Substitue in the point (3,1)
1 = a(9) +b(3) +c
Do the same for (1,9) and then again for (5,9)
You will have 3 equations and 3 variables. Solve for a, b, c.
There is also a formula to find the equation using the vertex and 1 point. you should be able to find it in your textbook or using a google search.
To determine the axis of symmetry of a quadratic equation, we need to find the x-coordinate of the vertex of the parabola. The vertex form of a quadratic equation is given by:
y = a(x - h)^2 + k,
where (h, k) represents the coordinates of the vertex. In this case, we are given two points that lie on the parabola: (3, 1) and (1, 9).
Step 1: Finding the equation of the parabola
Let's use the point (3, 1) to substitute into the vertex form equation:
1 = a(3 - h)^2 + k.
Step 2: Substitute the other point into the equation
Using the second point (1, 9), we can substitute it into the equation as well:
9 = a(1 - h)^2 + k.
Step 3: Set up a system of equations
We now have the following system of equations:
1 = a(3 - h)^2 + k,
9 = a(1 - h)^2 + k.
Step 4: Solve the system of equations
To solve the system of equations, we can subtract the first equation from the second equation:
9 - 1 = a(1 - h)^2 + k - (a(3 - h)^2 + k).
This simplifies to:
8 = a(1 - h)^2 - a(3 - h)^2,
8 = a[(1 - h)^2 - (3 - h)^2].
Step 5: Simplify and solve for a
Let's simplify the equation further:
8 = a[1 - 2h + h^2 - 9 + 6h - h^2],
8 = a (-8h - 8).
Dividing both sides by -8:
-1 = a(h + 1).
Step 6: Determine the value of h
Since the axis of symmetry of a parabola is the vertical line x = h, we can set h equal to 3 since we are given that the axis of symmetry is x = 3. Therefore, we have:
-1 = a(3 + 1),
-1 = a(4).
Dividing both sides by 4:
-1/4 = a.
So, we have found that a = -1/4.
Step 7: Substitute a into one of the original equations
Finally, substitute -1/4 for a in one of the original equations. Let's use the equation where (3, 1) is substituted:
1 = (-1/4)(3 - h)^2 + k.
We can simplify this equation further, but without additional information, we cannot find the exact values of h and k.