Suppose the graph of [tex]y=x^{\frac{5}{2}}[/tex] is shifted to the left 7 units. What is the equation that gives the new graph?
The equation that gives the new graph is [tex]y=[/tex] $\square$
Answer (1)
The original function is y = x 2 5 .
To shift the graph to the left by 7 units, replace x with ( x + 7 ) .
The new equation is y = ( x + 7 ) 2 5 .
The equation that gives the new graph is y = ( x + 7 ) 2 5 .
Explanation
Understanding the Problem The problem asks us to find the equation of a new graph after shifting the graph of y = x 2 5 to the left by 7 units. This is a transformation of functions problem.
Shifting the Graph To shift the graph of a function y = f ( x ) to the left by c units, we replace x with ( x + c ) in the function's equation. In this case, our original function is f ( x ) = x 2 5 , and we want to shift it to the left by c = 7 units.
Finding the New Equation So, we replace x with ( x + 7 ) in the equation y = x 2 5 . This gives us the new equation: y = ( x + 7 ) 2 5 .
Final Answer Therefore, the equation that gives the new graph is y = ( x + 7 ) 2 5 .
Examples
Imagine you are designing a video game where the character's jumping height is determined by the equation y = x 2 5 , where x is the time the jump button is pressed. If you want to make the character jump higher for the same amount of button press time, you can shift the graph to the left by changing the equation to y = ( x + 7 ) 2 5 . This means the character will reach a higher point sooner, making the game more exciting.
