Shrink the graph of f(x)=x+5 horizontally by a factor of 1/2
The Correct Answer and Explanation is :
To shrink the graph of ( f(x) = x + 5 ) horizontally by a factor of ( \frac{1}{2} ), we need to modify the function’s equation in a way that compresses it along the x-axis. A horizontal shrink is achieved by altering the argument of the function, specifically by multiplying ( x ) by a constant.
Step-by-step Explanation:
- Understanding Horizontal Shrink:
A horizontal shrink by a factor of ( \frac{1}{2} ) means that the function will be compressed so that the x-values will need to increase more rapidly in order to achieve the same output. For any general function ( f(x) ), if we apply a horizontal shrink by a factor of ( k ), the new function will be ( f(kx) ), where ( k > 1 ) results in a shrink. In this case, the factor is ( \frac{1}{2} ), so the transformation will be applied by replacing ( x ) with ( 2x ). This is because a horizontal shrink by ( \frac{1}{2} ) is the same as scaling ( x ) by ( 2 ), causing the graph to compress towards the y-axis. - Applying the Transformation:
The original function is:
[
f(x) = x + 5
]
After applying the horizontal shrink by a factor of ( \frac{1}{2} ), we replace ( x ) with ( 2x ). Thus, the transformed function is:
[
f(2x) = 2x + 5
] - Graphical Interpretation:
- Before the transformation, the graph of ( f(x) = x + 5 ) is a straight line with slope ( 1 ) and y-intercept ( 5 ).
- After applying the horizontal shrink, the slope of the new function remains ( 2 ), but the graph of the line is now closer to the y-axis. This means that for every unit change in ( y ), ( x ) changes more rapidly compared to the original graph.
Conclusion:
The graph of ( f(x) = x + 5 ), after shrinking horizontally by a factor of ( \frac{1}{2} ), becomes ( f(x) = 2x + 5 ). This transformation causes the graph to compress, and the x-values need to change more to achieve the same output values for ( y ).