To convert the parametric equations given by
[tex]\[
\begin{array}{l}
y = 2t^2 + 6 \\
x = 5t - 8
\end{array}
\][/tex]
into rectangular form [tex]\( y = f(x) \)[/tex], follow these steps:
1. Express [tex]\( t \)[/tex] in terms of [tex]\( x \)[/tex] from the second equation [tex]\( x = 5t - 8 \)[/tex]:
[tex]\[
x = 5t - 8
\][/tex]
Add 8 to both sides:
[tex]\[
x + 8 = 5t
\][/tex]
Now, divide by 5 to isolate [tex]\( t \)[/tex]:
[tex]\[
t = \frac{x + 8}{5}
\][/tex]
2. Substitute [tex]\( t = \frac{x + 8}{5} \)[/tex] into the first equation [tex]\( y = 2t^2 + 6 \)[/tex]:
[tex]\[
y = 2 \left( \frac{x + 8}{5} \right)^2 + 6
\][/tex]
3. Simplify the expression:
First, square the term inside parentheses:
[tex]\[
\left( \frac{x + 8}{5} \right)^2 = \frac{(x + 8)^2}{25}
\][/tex]
So, substituting back we get:
[tex]\[
y = 2 \cdot \frac{(x + 8)^2}{25} + 6
\][/tex]
Multiply 2 through the fraction:
[tex]\[
y = \frac{2(x + 8)^2}{25} + 6
\][/tex]
4. Expand the quadratic term [tex]\((x + 8)^2\)[/tex]:
[tex]\[
(x + 8)^2 = x^2 + 16x + 64
\][/tex]
Now substitute the expanded form back into the equation:
[tex]\[
y = \frac{2(x^2 + 16x + 64)}{25} + 6
\][/tex]
5. Distribute the 2 in the fraction:
[tex]\[
y = \frac{2x^2 + 32x + 128}{25} + 6
\][/tex]
6. Combine constants:
First, express 6 as a fraction with denominator 25 to add them together:
[tex]\[
6 = \frac{150}{25}
\][/tex]
Thus, combining the terms yields:
[tex]\[
y = \frac{2x^2 + 32x + 128 + 150}{25}
\][/tex]
Combine the constants in the numerator:
[tex]\[
y = \frac{2x^2 + 32x + 278}{25}
\][/tex]
Therefore, the rectangular form of the parametric equations is:
[tex]\[
\boxed{y = \frac{2x^2 + 32x + 278}{25}}
\][/tex]
