Question

Select the correct answer from the drop-down menu.The function f is given by the table of values as shown below.X-1.2.3.4.5f(x)-13.19.37.91.253Use the given table to complete the statements.The parent function of the function represented in the table is ___(quadratic,linear,exponential)If function f was translated down 4 units,the ___(f(x),x- and f(x), x) -values would be ___(increased by 4, multiplied by 4, divided by 4, decreased by 4)A point in the table for the transformed function would be ___ ((2,23),(1,52),(3,29),(4,87)

Answer 1

EXPLANATION:

To determine the parent function for this equation, we will begin by observing the relationship between the variables.

Note that the difference between the input values x, is constant. However, the difference between the output values is not constant. Therefore, we know that this is not a linear function.

We shall attempt to derive a quadratic equation and we'll start with the first 3 ordered pairs;

$(1,13),(2,19),(3,37)$

The general formula for a quadratic function is;

$y=ax^2+bx+c$

We will now substitute the first ordered pair into this equation and we'll have;

$\begin{gathered} 13=a(1)^2+b(1)+c \\ 13=a+b+c---(1) \end{gathered}$

We will substitute also the second ordered pair and we'll have;

$\begin{gathered} 19=a(2)^2+b(2)+c \\ 19=4a+2b+c---(2) \end{gathered}$

We go on to the third pair now;

$\begin{gathered} 37=a(3)^2+b(3)+c \\ 37=9a+3b+c---(3) \end{gathered}$

From equation (1), we will make c the subject of the equation;

$c=13-a-b$

Substitute any ordered pair and the value of c into the general equation;

$\begin{gathered} Take\text{ }(2,19) \\ 19=a(2)^2+b(2)+(13-a-b) \end{gathered}$$\begin{gathered} 19=4a+2b+13-a-b \\ 19=3a+b+13 \\ 19-13=3a+b \\ 6=3a+b---(i) \end{gathered}$

Substitute another ordered pair and the value of c into the general equation;

$\begin{gathered} Take\text{ }(3,37) \\ 37=a(3)^2+b(3)+(13-a-b) \\ 37=9a+3b+13-a-b \\ 37-13=8a+2b \\ 24=8a+2b---(ii) \end{gathered}$

We shall now solve equations (i) and (ii).

$\begin{gathered} In\text{ equation }(i): \\ b=6-3a \end{gathered}$

Substitute into equation (ii);

$\begin{gathered} 24=8a+2b \\ 24=8a+2(6-3a) \\ 24=8a+12-6a \\ 24-12=8a-6a \\ 12=2a \\ a=6 \end{gathered}$

Substitute into equation (i) and solve for b;

$\begin{gathered} 6=3a+b \\ 6=3(6)+b \\ 6=18+b \\ 6-18=b \\ b=-12 \end{gathered}$

We can now input the values of a and b into equation (1);

$\begin{gathered} 13=a+b+c \\ 13=6+(-12)+c \\ 13=6-12+c \\ 13=-6+c \\ 13+6=c \\ c=19 \end{gathered}$

We can substitute the values of all three variables back into the general equation;

$f(x)=ax^2+bx+c$$f(x)=6x^2-12x+19$

If the function is translated down 4 units we would have;

$\begin{gathered} f(x)=6x^2-12x+19-4 \\ f(x)=6x^2-12x+15 \end{gathered}$

Note that the values of f(x) would be decreased by 4 as shown above.

ANSWER:

$\begin{gathered} Parent\text{ }function: \\ f(x)=6x^2-12x+13 \end{gathered}$$f(x),\text{ }would\text{ }be\text{ }decreased\text{ }by\text{ }4$

To determine a point on the equation after the transformation, we would substitute the x value and it would result in the y value as shown in the ordered pair.