Question

The graphs below have the same shape. What is the equation of the red graph?

Answer 1

Hello!

The answer is:

b.
`g(x)=x^(2) +3`

Why?

To find what's the equation of the red graph, first, we need to find a function that intercepts the y-axis at 3 and its vertex is located at (0,3)

We must remember that the constant value in a quadratic equation (c) will tell us where the function intercepts the y-axis, so, we are looking for an equation which has a constant value equal to 3, it discards the options a, c and d, so the correct option would be the option b.

Option b function:

`g(x)=x^(2) +3`

Where,

`a=1\\b=0\\c=3`

Let's check if the option b is the correct option:

First, let's find the axis intercepts:

Y-axis intercept

`f(x)=x^(2)+3\\f(0)=0^(2)+3\\y=3`

X-axis intercept

`0=x^(2)+3\\x=√(-3)`

There is no x-axis intercepts since the square roots of negative numbers does not exists in the real numbers.

Finding the vertex coordinates:

The x coordinate of the vertex can be found using the following equation:

`x=-(b)/(2a)=-(0)/(2*1)=0`

The x coordinate of the vertex is located at x=0, so to find the y-coordinate we must substitute the x value into the parabola equation, so

`y=x^(2)+3=0^(2)+3=3`

So, the y coordinate of the vertex is located at y=3

Therefore, the vertex of the function is (0,3)

Hence, the calculated values of the option b match with the given graph.

The correct option is: b.
`g(x)=x^(2) +3`

Have a nice day!