Hello!
The answer is:
b. [tex]g(x)=x^{2} +3[/tex]
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:
[tex]g(x)=x^{2} +3[/tex]
Where,
[tex]a=1\\b=0\\c=3[/tex]
Let's check if the option b is the correct option:
First, let's find the axis intercepts:
Y-axis intercept
[tex]f(x)=x^{2}+3\\f(0)=0^{2}+3\\y=3[/tex]
X-axis intercept
[tex]0=x^{2}+3\\x=\sqrt{-3}[/tex]
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:
[tex]x=-\frac{b}{2a}=-\frac{0}{2*1}=0[/tex]
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
[tex]y=x^{2}+3=0^{2}+3=3[/tex]
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. [tex]g(x)=x^{2} +3[/tex]
Have a nice day!
