Question Find the equation of a parabola with vertex at $(-1,-3)$ passing through $(3,-1)$ and $(0,-4)$ .

Studdy Solution

STEP 1

Assumptions
1. The vertex form of a parabola is either $y=a(x-h)^2+k$ or $x=a(y-k)^2+h$ .
2. The vertex of the parabola is given as $(-1,-3)$ .
3. The parabola passes through the points $(3,-1)$ and $(0,-4)$ .
4. We need to determine the value of $a$ and decide which vertex form to use based on the orientation of the parabola.

STEP 2

Since we are given the vertex, we will start with the vertex form of the parabola. We will use the form $y=a(x-h)^2+k$ because the vertex is given in the form of $(h, k)$ which corresponds to the x-coordinate and y-coordinate respectively.

STEP 3

Substitute the vertex $(-1,-3)$ into the vertex form to get the equation of the parabola without the value of $a$ .
$y=a(x-(-1))^2+(-3)$

STEP 4

Simplify the equation by removing the parentheses around $-1$ and $-3$ .
$y=a(x+1)^2-3$

STEP 5

Now we need to find the value of $a$ . To do this, we will use one of the points that the parabola passes through, such as $(3,-1)$ .

STEP 6

Substitute the point $(3,-1)$ into the equation $y=a(x+1)^2-3$ to find $a$ .
$-1=a(3+1)^2-3$

STEP 7

Simplify the equation by performing the operations inside the parentheses.
$-1=a(4)^2-3$

STEP 8

Simplify the equation further by squaring 4.
$-1=16a-3$

STEP 9

Add 3 to both sides of the equation to isolate the term with $a$ .
$-1+3=16a-3+3$

STEP 10

Simplify the left side of the equation.
$2=16a$

STEP 11

Divide both sides of the equation by 16 to solve for $a$ .
$\frac{2}{16}=a$

STEP 12

Reduce the fraction to its simplest form.
$a=\frac{1}{8}$

STEP 13

Now that we have the value of $a$ , we can write the final equation of the parabola using the vertex form.
$y=\frac{1}{8}(x+1)^2-3$

STEP 14

To ensure that our value of $a$ is correct, we should verify it by substituting the other given point $(0,-4)$ into the equation and checking if the equation is satisfied.
$-4=\frac{1}{8}(0+1)^2-3$

STEP 15

Simplify the equation by performing the operations inside the parentheses.
$-4=\frac{1}{8}(1)^2-3$

STEP 16

Simplify the equation further by squaring 1.
$-4=\frac{1}{8}-3$

STEP 17

Convert 3 into a fraction with a common denominator of 8 to combine with $\frac{1}{8}$ .
$-4=\frac{1}{8}-\frac{24}{8}$

STEP 18

Subtract the fractions.
$-4=-\frac{23}{8}$

STEP 19

Since $-\frac{23}{8}$ does not equal $-4$ , there is a mistake in our calculation. We must go back and check our steps.

STEP 20

Re-evaluate the substitution of the point $(0,-4)$ into the equation.
$-4=\frac{1}{8}(0+1)^2-3$

STEP 21

Simplify the equation by performing the operations inside the parentheses.
$-4=\frac{1}{8}(1)^2-3$

STEP 22

Simplify the equation further by squaring 1.
$-4=\frac{1}{8}-3$

STEP 23

Convert 3 into a fraction with a common denominator of 8 to combine with $\frac{1}{8}$ .
$-4=\frac{1}{8}-\frac{24}{8}$

STEP 24

Subtract the fractions.
$-4=-\frac{23}{8}$

STEP 25

Realize that the previous steps were correct and that the mistake was in the assumption that $-\frac{23}{8}$ should equal $-4$ . In fact, $-\frac{23}{8}$ is equal to $-2.875$ , which is not equal to $-4$ . This indicates that our value of $a$ is incorrect, and we need to re-evaluate our calculation of $a$ using the point $(3,-1)$ .

STEP 26

Revisit the calculation of $a$ using the point $(3,-1)$ and the equation $y=a(x+1)^2-3$ .
$-1=a(3+1)^2-3$

STEP 27

Simplify the equation by performing the operations inside the parentheses.
$-1=a(4)^2-3$

STEP 28

Simplify the equation further by squaring 4.
$-1=16a-3$

STEP 29

Add 3 to both sides of the equation to isolate the term with $a$ .
$2=16a$

STEP 30

Divide both sides of the equation by 16 to solve for $a$ .
$a=\frac{2}{16}$

STEP 31

Reduce the fraction to its simplest form.
$a=\frac{1}{8}$

STEP 32

Realize that the calculation of $a$ was correct. The error must be in the verification step with the point $(0,-4)$ . Let's try verifying with the point $(0,-4)$ again.
$-4=\frac{1}{8}(0+1)^2-3$

STEP 33

Simplify the equation by performing the operations inside the parentheses.
$-4=\frac{1}{8}(1)^2-3$

STEP 34

Simplify the equation further by squaring 1.
$-4=\frac{1}{8}-3$

STEP 35

Convert 3 into a fraction with a common denominator of 8 to combine with $\frac{1}{8}$ .
$-4=\frac{1}{8}-\frac{24}{8}$

STEP 36

Subtract the fractions.
$-4=-\frac{23}{8}$

STEP 37

Realize that the error is not in the calculation, but in the assumption that the parabola is vertical. Since the parabola passes through $(0,-4)$ , which has the same y-coordinate as the vertex, the parabola must be horizontal. Therefore, we should use the form $x=a(y-k)^2+h$ .

STEP 38

Substitute the vertex $(-1,-3)$ into the horizontal vertex form to get the equation of the parabola without the value of $a$ .
$x=a(y-(-3))^2+(-1)$

STEP 39

Simplify the equation by removing the parentheses around $-3$ and $-1$ .
$x=a(y+3)^2-1$

STEP 40

Now we need to find the value of $a$ again, using one of the points that the parabola passes through, such as $(3,-1)$ .

STEP 41

Substitute the point $(3,-1)$ into the equation $x=a(y+3)^2-1$ to find $a$ .
$3=a(-1+3)^2-1$

STEP 42

Simplify the equation by performing the operations inside the parentheses.
$3=a(2)^2-1$

STEP 43

Simplify the equation further by squaring 2.
$3=4a-1$

STEP 44

Add 1 to both sides of the equation to isolate the term with $a$ .
$3+1=4a-1+1$

STEP 45

Simplify the left side of the equation.
$4=4a$

STEP 46

Divide both sides of the equation by 4 to solve for $a$ .
$a=\frac{4}{4}$

STEP 47

Simplify the fraction to get the value of $a$ .
$a=1$

STEP 48

Now that we have the correct value of $a$ , we can write the final equation of the parabola using the horizontal vertex form.
$x=(y+3)^2-1$

STEP 49

To ensure that our value of $a$ is correct, we should verify it by substituting the other given point $(0,-4)$ into the equation and checking if the equation is satisfied.
$0=(y+3)^2-1$

STEP 50

Substitute $y=-4$ into the equation.
$0=(-4+3)^2-1$

STEP 51

Simplify the equation by performing the operations inside the parentheses.
$0=(-1)^2-1$

STEP 52

Simplify the equation further by squaring -1.
$0=1-1$

STEP 53

Simplify the equation.
$0=0$

STEP 54

Since the equation is satisfied with the point $(0,-4)$ , our final equation of the parabola is correct.
The final equation of the parabola is:
$x=(y+3)^2-1$

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