Question

I need help graphing the points and to make sure my table and answer is corrrct :( !

Answer 1

Let's start by graphing the line function, g(x). We know it is a line because it don't have a quadratic term (like x²). To graph a line, we need only two points. The easies ones are for x = 0 and g(x) = 0.

x = 0:

`\begin{gathered} g(0)=(1)/(2)\cdot0+2 \\ g(0)=2 \end{gathered}`

g(x) = 0:

`\begin{gathered} 0=(1)/(2)x+2 \\ (1)/(2)x=-2 \\ x=-4 \end{gathered}`

So we get the table:

x | g(x)

0 | 2

-4 | 0

We put the points and draw a line connecting them, this is the line (graph of g(x)):

Now we need to draw f(x), which is a parabola because it is a second degree equation (quadratic equation).

It is important to get the coordinates of the vertex. We know that the x of the vertex is:

`\begin{gathered} x_v=-(b)/(2a) \\ x_v=-(0)/(2(-1)) \\ x_v=0 \end{gathered}`

To get the y value of the vertex, we input the x into the function:

`\begin{gathered} f(x_v)=f(0)=-0^2+7 \\ f(0)=7 \end{gathered}`

So the vertex has the coordinates (0,7).

It is also good to know where the parabola crosses the x-axis, which means x = 0, the roots of the function:

`\begin{gathered} f(x)=0 \\ -x^2+7=0 \\ x^2=7 \\ x=\pm\sqrt[]{7}\approx\pm2.6 \end{gathered}`

So the table we get is:

x | f(x)

0 | 7

-2.6 | 0

2.6 | 0

Putting these points, we can draw the parabola.

However, you can also get more points. In this case we also want to solve the system of these functions. So we want coordinates where those graphs interscet each other. One way we can use to find it is to make:

`\begin{gathered} g(x)=f(x) \\ (1)/(2)x+2=-x^2+7 \\ x^2+(1)/(2)x+2-7=0 \\ x^2+(1)/(2)x-5=0 \end{gathered}`

And we can use Bharkara formula to get the values of x:

`\begin{gathered} x=\frac{-(1)/(2)\pm\sqrt[]{(1)/(4)-4\cdot1\cdot(-5)}}{2\cdot1} \\ x=\frac{-(1)/(2)\pm\sqrt[]{(1)/(4)+20}}{2} \\ x=\frac{-(1)/(2)\pm\sqrt[]{(81)/(4)}}{2} \\ x=(-(1)/(2)\pm(9)/(2))/(2) \\ x=(-1\pm9)/(4) \\ x_1=(-1+9)/(4)=(8)/(4)=2 \\ x_2=(-1-9)/(4)=-(10)/(4)=-2.5 \end{gathered}`

And then we get the y value by substituting either in f(x) or in g(x):

`\begin{gathered} g(2)=(1)/(2)\cdot2+2=3 \\ g(-2.5)=(1)/(2)\cdot(-2.5)+2=-1.25+2=0.75 \end{gathered}`

So, the intersections are in (-2.5, 0.75) and (2,3). And so we draw the graphs together:

The system of equations have two solutions, but the question asks for only one of them, which can only be (2,3), since there is no alternative for the other solution.