aksed find the equation oft he parabola y=ax^2 + bx +c. there are four given clues
1) the y-intercept is (0,6)
2) the curve goes throuth (4,5)
3) the curve has aturning point at (2,3)
4) the line of symmetry is x=1
a)is it not possible for all four clues to be true together (why not?)
b)which combinations of clues enable you to find a set of values a,b and c?
c)find the resulting parabola for each successful combination
7+4+5-3+9-10+7-8
(a) If the curve has a "turning point" (minimum or maximum) at (2,3), its line of symmetry would have to be x=2.
To find the equation of a parabola in the form y = ax^2 + bx + c, we usually need three pieces of information. However, if there are additional conditions such as the line of symmetry or certain points that the curve passes through, we may be able to solve for a, b, and c using all four given clues.
Let's analyze each clue and see which combinations of clues would allow us to find the values of a, b, and c.
1) The y-intercept is (0,6).
The y-intercept occurs when x = 0. Plugging this into the equation, we have y = a(0)^2 + b(0) + c = c. Therefore, the value of c is 6. This clue tells us the value of c.
2) The curve goes through (4,5).
Plugging x = 4 and y = 5 into the equation, we have 5 = a(4)^2 + b(4) + 6. Simplifying this equation will allow us to solve for both a and b. This clue helps us find the values of a and b.
3) The curve has a turning point at (2,3).
The turning point of a parabola occurs when x = h, where h is the x-coordinate of the turning point. In this case, h = 2. The formula for the x-coordinate of the turning point is given by x = -b / (2a). Plugging in the values we have, we get 2 = -b / (2a). This equation, combined with the previous information, will enable us to solve for the remaining unknown values. This clue helps us find the values of a and b.
4) The line of symmetry is x = 1.
The line of symmetry of a parabola occurs when x = h, the x-coordinate of the turning point. From the previous clue, we know that h = 2. Therefore, the line of symmetry is x = 2. If the line of symmetry is given as x = p, then the x-coordinate of the turning point is also p. This clue simply confirms the x-coordinate of the turning point, which we already know.
Now, let's answer the questions:
a) It is possible for all four clues to be true together. Each clue provides valuable information about the parabola, allowing us to find the equation.
b) The combinations of clues that enable us to find the values of a, b, and c are 1, 2, and 3. These clues provide enough information to solve for all three unknowns.
c) To find the resulting parabola, we need to plug in the values of a, b, and c into the equation y = ax^2 + bx + c.
Using the information we obtained:
a) a and b can be found using clues 2 and 3.
b) c is given in clue 1.
c) a = -1, b = 3, and c = 6.
Therefore, the resulting parabola is y = -x^2 + 3x + 6.