Solved on Jan 21, 2024

Find the equation of a parabola given its vertex at $(1, -3)$ and a point it passes through at $(0, \frac{7}{2})$ .

STEP 1

Assumptions
1. The general form of a parabola with vertex $(h, k)$ is $y = a(x - h)^2 + k$ .
2. The vertex of the parabola is given as $(1, -3)$ .
3. The parabola passes through the point $(0, \frac{7}{2})$ .

STEP 2

We will use the vertex form of the parabola to write the equation.
$y = a(x - h)^2 + k$

STEP 3

Substitute the vertex $(h, k)$ into the vertex form equation.
$y = a(x - 1)^2 - 3$

STEP 4

Now we need to find the value of $a$ . We can do this by using the point $(0, \frac{7}{2})$ that the parabola passes through.

STEP 5

Substitute the point $(0, \frac{7}{2})$ into the equation to solve for $a$ .
$\frac{7}{2} = a(0 - 1)^2 - 3$

STEP 6

Simplify the equation by squaring the term $(0 - 1)^2$ .
$\frac{7}{2} = a(1) - 3$

STEP 7

Add $3$ to both sides of the equation to isolate $a$ .
$\frac{7}{2} + 3 = a$

STEP 8

Convert $3$ to a fraction with the same denominator as $\frac{7}{2}$ to combine the terms.
$3 = \frac{6}{2}$
$\frac{7}{2} + \frac{6}{2} = a$

STEP 9

Add the fractions to solve for $a$ .
$a = \frac{7}{2} + \frac{6}{2} = \frac{13}{2}$

STEP 10

Now that we have the value of $a$ , we can write the final equation of the parabola.
$y = \frac{13}{2}(x - 1)^2 - 3$
This is the equation of the parabola with the given vertex and passing through the given point.

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Find the equation of a parabola given its vertex at $(1, -3)$ and a point it passes through at $(0, \frac{7}{2})$ .

STEP 1

Assumptions
1. The general form of a parabola with vertex $(h, k)$ is $y = a(x - h)^2 + k$ .
2. The vertex of the parabola is given as $(1, -3)$ .
3. The parabola passes through the point $(0, \frac{7}{2})$ .

STEP 2

We will use the vertex form of the parabola to write the equation.
$y = a(x - h)^2 + k$

STEP 3

Substitute the vertex $(h, k)$ into the vertex form equation.
$y = a(x - 1)^2 - 3$

STEP 4

Now we need to find the value of $a$ . We can do this by using the point $(0, \frac{7}{2})$ that the parabola passes through.

STEP 5

Substitute the point $(0, \frac{7}{2})$ into the equation to solve for $a$ .
$\frac{7}{2} = a(0 - 1)^2 - 3$

STEP 6

Simplify the equation by squaring the term $(0 - 1)^2$ .
$\frac{7}{2} = a(1) - 3$

STEP 7

Add $3$ to both sides of the equation to isolate $a$ .
$\frac{7}{2} + 3 = a$

STEP 8

Convert $3$ to a fraction with the same denominator as $\frac{7}{2}$ to combine the terms.
$3 = \frac{6}{2}$
$\frac{7}{2} + \frac{6}{2} = a$

STEP 9

Add the fractions to solve for $a$ .
$a = \frac{7}{2} + \frac{6}{2} = \frac{13}{2}$

STEP 10

Now that we have the value of $a$ , we can write the final equation of the parabola.
$y = \frac{13}{2}(x - 1)^2 - 3$
This is the equation of the parabola with the given vertex and passing through the given point.

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Find the equation of a parabola given its vertex at (1,−3)(1, -3)(1,−3) and a point it passes through at (0,72)(0, \frac{7}{2})(0,27​).

STEP 1

Assumptions1. The general form of a parabola with vertex (h,k)(h, k)(h,k) is y=a(x−h)2+ky = a(x - h)^2 + ky=a(x−h)2+k.2. The vertex of the parabola is given as (1,−3)(1, -3)(1,−3).3. The parabola passes through the point (0,72)(0, \frac{7}{2})(0,27​).

STEP 2

We will use the vertex form of the parabola to write the equation.y=a(x−h)2+ky = a(x - h)^2 + ky=a(x−h)2+k

STEP 3

Substitute the vertex (h,k)(h, k)(h,k) into the vertex form equation.y=a(x−1)2−3y = a(x - 1)^2 - 3y=a(x−1)2−3

STEP 4

Now we need to find the value of aaa. We can do this by using the point (0,72)(0, \frac{7}{2})(0,27​) that the parabola passes through.

STEP 5

Substitute the point (0,72)(0, \frac{7}{2})(0,27​) into the equation to solve for aaa.72=a(0−1)2−3\frac{7}{2} = a(0 - 1)^2 - 327​=a(0−1)2−3

STEP 6

Simplify the equation by squaring the term (0−1)2(0 - 1)^2(0−1)2.72=a(1)−3\frac{7}{2} = a(1) - 327​=a(1)−3

STEP 7

Add 333 to both sides of the equation to isolate aaa.72+3=a\frac{7}{2} + 3 = a27​+3=a

STEP 8

Convert 333 to a fraction with the same denominator as 72\frac{7}{2}27​ to combine the terms.3=623 = \frac{6}{2}3=26​72+62=a\frac{7}{2} + \frac{6}{2} = a27​+26​=a

STEP 9

Add the fractions to solve for aaa.a=72+62=132a = \frac{7}{2} + \frac{6}{2} = \frac{13}{2}a=27​+26​=213​

STEP 10

Now that we have the value of aaa, we can write the final equation of the parabola.y=132(x−1)2−3y = \frac{13}{2}(x - 1)^2 - 3y=213​(x−1)2−3This is the equation of the parabola with the given vertex and passing through the given point.

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

Find the equation of a parabola given its vertex at $(1, -3)$ and a point it passes through at $(0, \frac{7}{2})$ .

Assumptions
1. The general form of a parabola with vertex $(h, k)$ is $y = a(x - h)^2 + k$ .
2. The vertex of the parabola is given as $(1, -3)$ .
3. The parabola passes through the point $(0, \frac{7}{2})$ .

We will use the vertex form of the parabola to write the equation.
$y = a(x - h)^2 + k$

Substitute the vertex $(h, k)$ into the vertex form equation.
$y = a(x - 1)^2 - 3$

Now we need to find the value of $a$ . We can do this by using the point $(0, \frac{7}{2})$ that the parabola passes through.

Substitute the point $(0, \frac{7}{2})$ into the equation to solve for $a$ .
$\frac{7}{2} = a(0 - 1)^2 - 3$

Simplify the equation by squaring the term $(0 - 1)^2$ .
$\frac{7}{2} = a(1) - 3$

Add $3$ to both sides of the equation to isolate $a$ .
$\frac{7}{2} + 3 = a$

Convert $3$ to a fraction with the same denominator as $\frac{7}{2}$ to combine the terms.
$3 = \frac{6}{2}$
$\frac{7}{2} + \frac{6}{2} = a$

Add the fractions to solve for $a$ .
$a = \frac{7}{2} + \frac{6}{2} = \frac{13}{2}$

Now that we have the value of $a$ , we can write the final equation of the parabola.
$y = \frac{13}{2}(x - 1)^2 - 3$
This is the equation of the parabola with the given vertex and passing through the given point.