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bezimeni [28]
2 years ago
14

Can someone please help me with my Question #29 of The Quadratic Relations for me please?

Mathematics
1 answer:
lara [203]2 years ago
7 0

Answer:

Calculate the <u>first differences</u> between the y-values:

\sf 3 \underset{+1}{\longrightarrow} 4 \underset{+3}{\longrightarrow} 7 \underset{+5}{\longrightarrow} 12 \underset{+7}{\longrightarrow} 19

As the first differences are <u>not the same</u>, we need to calculate the <u>second differences</u>:

\sf 1 \underset{+2}{\longrightarrow} 3 \underset{+2}{\longrightarrow} 5 \underset{+2}{\longrightarrow} 7

As the second differences are the <u>same</u>, the relationship between the variable is quadratic and will contain an x^2  term.

--------------------------------------------------------------------------------------------------

<u>To determine the quadratic equation</u>

The coefficient of x^2  is always <u>half</u> of the <u>second difference</u>.

As the second difference is 2, and half of 2 is 1, the coefficient of x^2 is 1.

The standard form of a quadratic equation is:  y=ax^2+bx+c

(where a, b and c are constants to be found).

We have already determined that the coefficient of x^2 is 1.

Therefore, a = 1

From the given table, when x=0, y=12.

\implies a(0)^2+b(0)+c=12

\implies c=12

Finally, to find b, substitute the found values of a and c into the equation, then substitute one of the ordered pairs from the given table:

\begin{aligned}\implies x^2+bx+12 & = y\\ \textsf{at }(1,19) \implies (1)^2+b(1)+12 & = 19\\ 1+b+12 & = 19\\b+13 & =19\\b&=6\end{aligned}

Therefore, the quadratic equation for the given ordered pairs is:

y=x^2+6x+12

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