Question

Write a quadratic function that fits the given set of points.

Answer 1

1. (-4, 9), (0, -7) and (1, -1)

2. (2, 3), (6, 3), and (8, -3)

3. (-1, -12), (1, 0), and (2, 9)

Answer 2

A quadratic function is:

\(f(x) = ax^2 + bx + c \)

Using one point above at a time in the equation above, find an equation in the variables a, b, and c. Then solve the three equations simultaneously to find a, b, and c. Then replace a, b, and c with the values you find.

Answer 3

That really doesn't make sense to me, can you explain it?

Answer 4

Like give me an example?

Answer 5

I'll show you how to get the first equation with the first set of points.

Answer 6

Take the form equation I gave you:

\( f(x) = ax^2 + bx + c\)

Now we are working on problem 1: (-4, 9), (0, -7) and (1, -1)

Let's use the first given point (-4, 9). We insert -4 for x and 9 for y (or f(x)).

\(9 = a(-4)^2 + x(-4) + c\)

Now we simplify this:

9 = 16a - 4b + c

Switch sides:

16a - 4b + c = 9

Now we have one equation in the variables a, b, and c.

Using the second and third points of problem 1, we come up with 2 more equations in variables a, b, and c. Now insert the second point, and come up with an equation. Then insert the third point and come up with an equation. Can you do that?

Then we solve the three equations as a system of equations to find a, b, and c.

Answer 7

so we have 3 equations:

9 = 16a + 4b + c
7 = c
-1 = a - b + c

Answer 8

@mathstudent55

Answer 9

For the second point, we have (0, -7):

\(a(0)^2 + b(0) + c = -7\)

or

c = -7

The third point gives us:

\(a(1)^2 + b(1) + c = -1\)

a + b + c = -1

Answer 10

First equation is
16a - 4b + c = 9

Answer 11

so 8 = 16 - 4 + 7 ?

Answer 12

Now we have a system of 3 equations:

16a - 4b + c = 9
c = -7
a + b + c = -1

Since we already know c, we substitute -7 for c in the first and third equations. Then we'll have simply a system of two equations.

Answer 13

16a - 4b -7 = 9
a + b -7 = -1

Which becomes:

16a - 4b = 16
a + b = 6

Answer 14

We can use substitution to solve the system.

From the second equation we get:
b = 6 - a

Now we substitute into the first equation:
16a - 4(6 - a) = 16

16a - 24 + 4a = 16
20a = 40
a = 2

Now we substitute a = 2 into b = 6 - a to find b:

b = 6 - 2
b = 4

Now we have a = 2, b = 4, and c = -7

Now we can write the quadratic function:

\(f(x) = 2x^2 + 4x - 7\)

Answer 15

With a little programming, Mathematica can solve the procedural part in one statement. Refer to the attachment.