Question

The graph of f(x) = 2x is shown on the grid. The graph of g(x) = ()x is the graph of f(x) = 2x reflected over the y-axis. Which graph represents g(x)?

Answer 1

`g(x)=-2x`

Explanation:

A point reflected across the y-axis maintains its y-coordinate, but its x-coordinate switches signs. So, a positive x-coordinate becomes negative, and a negative x-coordinate becomes positive.

Let's take a few points from the original function, f(x). Remember, if we know the function, we can find the y-coordinate for any x-coordiante by simply plugging it into the function's equation.

Generally,
`f(x)=2x`

So:

`f(0)=2(0)=0\\f(1)=2(1)=2\\f(2)=2(2)=4`

Leading us to have the plot points (0,0), (1,2) and (2,4).

To reflect this across the y-axis for the g(x) equation, we just need to turn the x-coordinates negative, resulting in a set of (0,0), (-1,2), and (-2,4).

Since we know this is a linear function (because there are no exponents in the equation), we can calculate the slope of this new set of points by using just 2 of them. The slope will give us our equation, because since (0,0) is a point on our line, we know that the y-intercept is zero.

`slope=((y_(2)-y_(1)))/((x_(2)-x_(1))) \\slope=((4-2))/(((-2)-(-1))) \\slope=(2)/(-1)\\slope=-2\\\\g(x)=-2(x)`