Find the quadratic function passing through the points , and .

Ask by Ingram Griffiths. in the United States

Mar 21,2025

Question

Find the quadratic function passing through the points $$ (-2,-13),(2,-17) $$, and $$ (1,-7) $$. $$ F(x)=\square x^{2}+\square x+\square $$

UpStudy AI · Accepted Answer

We start with the general quadratic function:
$$
F(x)=ax^2+bx+c.
$$

Given the points, we write the following equations:

1. For $$(-2,-13)$$:
$$
4a - 2b + c = -13.
$$

2. For $$(2,-17)$$:
$$
4a + 2b + c = -17.
$$

3. For $$(1,-7)$$:
$$
a + b + c = -7.
$$

Subtract the first equation from the second:
$$
(4a+2b+c) - (4a-2b+c)=4b=-17-(-13)=-4,
$$
so:
$$
b=-1.
$$

Substitute $$ b=-1 $$ into the third equation:
$$
a - 1 + c = -7 \quad \Longrightarrow \quad a + c = -6 \quad \Longrightarrow \quad c = -6 - a.
$$

Now substitute $$ b=-1 $$ and $$ c=-6-a $$ in the first equation:
$$
4a - 2(-1) + (-6-a) = -13.
$$
Simplify:
$$
4a + 2 - 6 - a = -13 \quad \Longrightarrow \quad 3a - 4 = -13.
$$
Solve for $$ a $$:
$$
3a = -9 \quad \Longrightarrow \quad a = -3.
$$

Then, substitute $$ a=-3 $$ into $$ c=-6-a $$:
$$
c=-6-(-3)=-3.
$$

Thus, the quadratic function is:
$$
F(x)=-3x^2- x-3.
$$
The quadratic function is $$ F(x) = -3x^2 - x - 3 $$.

Find the quadratic function passing through the points , and .

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