Solved on Jan 12, 2024

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

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 $y = -2x^2 + 16x - 25$ .

Was this helpful?

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

STEP 1

Assumptions1. The general equation of a parabola is y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c.2. The parabola passes through the points (3,5)(3,5)(3,5), (4,7)(4,7)(4,7), and (5,5)(5,5)(5,5).3. We need to find the coefficients aaa, bbb, and ccc that satisfy the given points.

STEP 2

Substitute the coordinates of the first point (3,5)(3,5)(3,5) into the general equation of the parabola.5=a(3)2+b(3)+c5 = a(3)^2 + b(3) + c5=a(3)2+b(3)+c

STEP 3

Simplify the equation from STEP_2.5=9a+3b+c5 = 9a + 3b + c5=9a+3b+c

STEP 4

Substitute the coordinates of the second point (4,7)(4,7)(4,7) into the general equation of the parabola.7=a(4)2+b(4)+c7 = a(4)^2 + b(4) + c7=a(4)2+b(4)+c

STEP 5

Simplify the equation from STEP_4.7=16a+4b+c7 = 16a + 4b + c7=16a+4b+c

STEP 6

Substitute the coordinates of the third point (5,5)(5,5)(5,5) into the general equation of the parabola.5=a(5)2+b(5)+c5 = a(5)^2 + b(5) + c5=a(5)2+b(5)+c

STEP 7

Simplify the equation from STEP_6.5=25a+5b+c5 = 25a + 5b + c5=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 ccc 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 bbb with the second equation.2(7a+b)=2(2)2(7a + b) = 2(2)2(7a+b)=2(2)

STEP 12

Simplify the equation from STEP_11.14a+2b=414a + 2b = 414a+2b=4

STEP 13

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

STEP 14

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

STEP 15

Solve for aaa.a=−42=−2a = -\frac{4}{2} = -2a=−24​=−2

STEP 16

Substitute the value of aaa into one of the equations from STEP_10 to find bbb.7(−2)+b=27(-2) + b = 27(−2)+b=2

STEP 17

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

STEP 18

Substitute the values of aaa and bbb into one of the original equations to find ccc.9(−2)+3(16)+c=59(-2) + 3(16) + c = 59(−2)+3(16)+c=5

STEP 19

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

STEP 20

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

STEP 21

Simplify the calculation for ccc.c=23−48c = 23 - 48c=23−48

STEP 22

Finish solving for ccc.c=−25c = -25c=−25

STEP 23

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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

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.

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

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

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

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

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

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

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}

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}

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

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

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

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

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

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

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

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

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

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

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

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

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