I was asked to deduce the values of a,b, and c for the parabola y= ax^2 + bx +c. There are 4 clues given:

1. the y-intercept is (0,6)
2. The curve goes through (4,5)
3. The curvev has a turning point at (2,3)
4. the line of symmetry is x=1
Questions: It is not possible for all clues to be true together (Why not?)
Which combinations of clues enable you to find a set of values for a,b and c?
Find the resulting parabola for each successful combination.
Tq..

To deduce the values of a, b, and c for the parabola y = ax^2 + bx + c, we need to use the given clues to form a system of equations.

1. The y-intercept is (0,6):
When x = 0, y = 6. Since the y-intercept is the point where the parabola crosses the y-axis, we can substitute these values into the equation:
6 = a(0)^2 + b(0) + c
This simplifies to c = 6.

2. The curve goes through (4,5):
Since the point (4,5) lies on the parabola, we can substitute these values into the equation:
5 = a(4)^2 + b(4) + 6
This equation allows us to find the relationship between a and b.

3. The curve has a turning point at (2,3):
The turning point of a parabola is also known as the vertex. The x-coordinate of the vertex is given as 2, which means that the line of symmetry is x = 2. Since the line of symmetry is x = 1 according to clue 4, it contradicts clue 3. This means that clues 3 and 4 cannot be true together.

4. The line of symmetry is x = 1:
The line of symmetry for a parabola in the form y = ax^2 + bx + c is given by x = -b / (2a). Setting this equal to 1, we get:
1 = -b / (2a)
This equation allows us to find the relationship between a and b.

Now, let's examine the combinations of clues that enable us to find a set of values for a, b, and c:

- Clues 1 and 2: With these two clues, we have two equations:
6 = c
5 = 16a + 4b + 6
From these equations, we can solve for a and b and substitute the values back into the equation to find c. This will give us a unique parabola.

- Clues 2 and 4: With these two clues, we have two equations:
5 = 16a + 4b + 6
1 = -b / (2a)
From these equations, we can solve for a and b, which will imply the parabolic equation. However, we cannot determine the value of c without additional information.

Therefore, only the combination of clues 1 and 2 allows us to find the resulting parabola fully.

To find the resulting parabola for clues 1 and 2:
Using the system of equations formed in the combination of clues 1 and 2, solve for a and b:

6 = c
5 = 16a + 4b + 6

From the second equation:
16a + 4b = -1

Now, we can substitute the value of c = 6 from clue 1 into the equation:
y = ax^2 + bx + c
This gives us the final parabolic equation:

y = ax^2 + bx + 6

By solving the system of equations, you can find the specific values of a and b to complete the equation.