Question Find the equation of the parabola that passes through the points $(3,5), (4,7)$ , and $(5,5)$ .

Studdy Solution

STEP 1

Assumptions
1. The general equation of a parabola is $y = ax^2 + bx + c$ .
2. The parabola passes through the points $(3,5)$ , $(4,7)$ , and $(5,5)$ .
3. We need to find the coefficients $a$ , $b$ , and $c$ that satisfy the given points.

STEP 2

Substitute the coordinates of the first point $(3,5)$ into the general equation of the parabola.
$5 = a(3)^2 + b(3) + c$

STEP 3

Simplify the equation from STEP_2.
$5 = 9a + 3b + c$

STEP 4

Substitute the coordinates of the second point $(4,7)$ into the general equation of the parabola.
$7 = a(4)^2 + b(4) + c$

STEP 5

Simplify the equation from STEP_4.
$7 = 16a + 4b + c$

STEP 6

Substitute the coordinates of the third point $(5,5)$ into the general equation of the parabola.
$5 = a(5)^2 + b(5) + c$

STEP 7

Simplify the equation from STEP_6.
$5 = 25a + 5b + c$

STEP 8

Now we have a system of three equations with three unknowns:
\begin{align} 9a + 3b + c &= 5 \\ 16a + 4b + c &= 7 \\ 25a + 5b + c &= 5 \end{align}

STEP 9

Subtract the first equation from the second and the third equations to eliminate $c$ and simplify the system.
\begin{align} (16a + 4b + c) - (9a + 3b + c) &= 7 - 5 \\ (25a + 5b + c) - (9a + 3b + c) &= 5 - 5 \end{align}

STEP 10

Simplify the equations from STEP_9.
\begin{align} 7a + b &= 2 \\ 16a + 2b &= 0 \end{align}

STEP 11

Multiply the first equation by 2 to align the coefficients of $b$ with the second equation.
$2(7a + b) = 2(2)$

STEP 12

Simplify the equation from STEP_11.
$14a + 2b = 4$

STEP 13

Subtract the second equation from the new equation obtained in STEP_12 to eliminate $b$ .
$(14a + 2b) - (16a + 2b) = 4 - 0$

STEP 14

Simplify the equation from STEP_13.
$-2a = 4$

STEP 15

Solve for $a$ .
$a = -\frac{4}{2} = -2$

STEP 16

Substitute the value of $a$ into one of the equations from STEP_10 to find $b$ .
$7(-2) + b = 2$

STEP 17

Solve for $b$ .
$b = 2 + 14 = 16$

STEP 18

Substitute the values of $a$ and $b$ into one of the original equations to find $c$ .
$9(-2) + 3(16) + c = 5$

STEP 19

Solve for $c$ .
$c = 5 - 9(-2) - 3(16)$

STEP 20

Calculate the value of $c$ .
$c = 5 + 18 - 48$

STEP 21

Simplify the calculation for $c$ .
$c = 23 - 48$

STEP 22

Finish solving for $c$ .
$c = -25$

STEP 23

Now that we have the values of $a$ , $b$ , and $c$ , we can write the equation of the parabola.
$y = -2x^2 + 16x - 25$
The equation of the parabola that passes through the points $(3,5)$ , $(4,7)$ , and $(5,5)$ is $y = -2x^2 + 16x - 25$ .

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