The graph represents the function f(x)=x2 +3x+2 .
If g(x) is the reflection of f(x) across the x-axis, g(x)= . (Write the function in standard form. Use ^ to indicate an exponent.)
The Correct Answer and Explanation is:
To find the function ( g(x) ), which is the reflection of ( f(x) = x^2 + 3x + 2 ) across the x-axis, we can use the property that reflecting a function across the x-axis involves multiplying the entire function by (-1). Thus, the transformation can be represented mathematically as follows:
[
g(x) = -f(x)
]
Substituting ( f(x) ) into this equation, we get:
[
g(x) = – (x^2 + 3x + 2)
]
Distributing the negative sign throughout the function gives:
[
g(x) = -x^2 – 3x – 2
]
This is the function of ( g(x) ) in standard form. The standard form of a quadratic function is typically expressed as:
[
g(x) = ax^2 + bx + c
]
In this case, we have ( a = -1 ), ( b = -3 ), and ( c = -2 ).
Explanation:
The function ( f(x) = x^2 + 3x + 2 ) represents a quadratic equation, which graphs as a parabola opening upwards. The reflection of this function across the x-axis will result in a parabola that opens downwards. The original function can be analyzed to find its vertex, intercepts, and general shape, but for the purposes of reflecting it, the most important aspect is understanding that each point on the graph of ( f(x) ) will have its y-coordinate inverted for ( g(x) ).
When we reflect across the x-axis, we change the sign of the output value. For instance, if ( f(a) = b ), then for the reflection, we have ( g(a) = -b ). This property helps us visualize how every point on the graph of ( f ) corresponds to a point on ( g ) directly below it, resulting in the inverted parabola.
Finally, the graph of ( g(x) ) will have the same x-intercepts as ( f(x) ) because the x-values where the function equals zero do not change during reflection. The original function has roots at ( x = -1 ) and ( x = -2 ), which will be preserved in the reflection. Thus, the reflection maintains the structure of the quadratic, but inverts it to open downwards.