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scoray [572]
3 years ago
11

Write the equation of the quadratic function whose graph passes through

ormula1" title="(-3,2)" alt="(-3,2)" align="absmiddle" class="latex-formula">, (-1,0), and (1,6).
Mathematics
1 answer:
blagie [28]3 years ago
5 0

Answer:

f(x)=x^2+3x+2

Step-by-step explanation:

We want to write the equation of a quadratic whose graph passes through (-3, 2), (-1, 0), and (1, 6).

Remember that the standard quadratic function is given by:

f(x)=ax^2+bx+c

Since it passes through the point (-3, 2). This means that when x=-3, f(x)=f(-3)=2. Hence:

f(-3)=2=a(-3)^2+b(-3)+c

Simplify:

2=9a-3b+c

Perform the same computations for the coordinates (-1, 0) and (1, 6). Therefore:

0=a(-1)^2+b(-1)+c \\ \\0=a-b+c

And for (1, 6):

6=a(1)^2+b(1)+c\\\\ 6=a+b+c

So, we have a triple system of equations:

\left\{        \begin{array}{ll}            2=9a-3b+c &\\           0=a-b+c \\6=a+b+c        \end{array}    \right.

We can solve this using elimination.

Notice that the b term in Equation 2 and 3 are opposites. Hence, let's add them together. This yields:

(0+6)=(a+a)+(-b+b)+(c+c)

Compute:

6=2a+2c

Let's divide both sides by 2:

3=a+c

Now, let's eliminate b again but we will use Equation 1 and 2.

Notice that if we multiply Equation 2 by -3, then the b terms will be opposites. So:

-3(0)=-3(a-b+c)

Multiply:

0=-3a+3b-3c

Add this to Equation 1:

(0+2)=(9a-3a)+(-3b+3b)+(c-3c)

Compute:

2=6a-2c

Again, we can divide both sides by 2:

1=3a-c

So, we know have two equations with only two variables:

3=a+c\text{ and } 1=3a-c

We can solve for a using elimination since the c term are opposites of each other. Add the two equations together:

(3+1)=(a+3a)+(c-c)

Compute:

4=4a

Solve for a:

a=1

So, the value of a is 1.

Using either of the two equations, we can now find c. Let's use the first one. Hence:

3=a+c

Substitute 1 for a and solve for c:

\begin{aligned} c+(1)&=3 \\c&=2 \end{aligned}

So, the value of c is 2.

Finally, using any of the three original equations, solve for b:

We can use Equation 3. Hence:

6=a+b+c

Substitute in known values and solve for b:

6=(1)+b+(2)\\\\6=3+b\\\\b=3

Therefore, a=1, b=3, and c=2.

Hence, our quadratic function is:

f(x)=x^2+3x+2

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Answer:

165 is greater than or equal to 17.75x+5.25; 9 is greater than or equal to x

Step-by-step explanation:

The problem states that they have no more than $165. This means that they either use all the money, or they have leftover money, which means you need to use the 'greater than or equal to' sign. It says that tickets are $17.75 a person, and we're trying to find the number of people (x). You can write this as 17.75x. There is also a parking fee to add on to the equation, which is represented as +5.25. Now, we can solve!

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