Mathematics

Question

the graph of y=√x is shifted 2 units up and 5 units left, Which equation represents the new graph?

A. y=-√x+5) +2

B. y=-√x-2) +5

C. y=-√x+2) -5

D. y=-√x-5) +2

2 Answer

  • Answer:

    A. [tex]y=\sqrt{x+5}+2[/tex].

    Step-by-step explanation:

    We are given the function [tex]y=\sqrt{x}[/tex].

    Now, the function is shifted 2 units up and 5 units to the left.

    That is, the function is translated 2 units up and 5 units to the left.

    Since, we know,

    Translation of 'k' units up changes the function [tex]f(x)[/tex] to [tex]f(x)+k[/tex].

    So, the function translated 2 units up is [tex]y=\sqrt{x}+2[/tex].

    Translation of 'k' units to the left changes the function [tex]f(x)[/tex] to [tex]f(x+k)[/tex].

    So, the new function translated 5 units left is [tex]y=\sqrt{x+5}+2[/tex].

    Hence, the equation representing the new function is [tex]y=\sqrt{x+5}+2[/tex].

  • y = √(x + 5) + 2

    Further explanation

    Given:

    The graph of [tex]y = \sqrt{x}[/tex] is

    • shifted 2 units up, and
    • 5 units left.

    Question:

    Which equation represents the new graph?

    The Process:

    The translation is a form of transformation geometry.

    Translation (or shifting): moving a graph on an analytic plane without changing its shape.

    In general, given the graph of y = f(x) and v > 0, we obtain the graph of:  

    • [tex]\boxed{ \ y = f(x) + v \ }[/tex] by shifting the graph of [tex]\boxed{ \ y = f(x) \ }[/tex] upward v units.  
    • [tex]\boxed{ \ y = f(x) - v \ }[/tex] by shifting the graph of [tex]\boxed{ \ y = f(x) \ }[/tex] downward v units.  

    That's the vertical shift, now the horizontal one. Given the graph of y = f(x) and h > 0, we obtain the graph of:  

    • [tex]\boxed{ \ y = f(x + h) \ }[/tex] by shifting the graph of [tex]\boxed{ \ y = f(x) \ }[/tex] to the left h units.  
    • [tex]\boxed{ \ y = f(x - h) \ }[/tex] by shifting the graph of [tex]\boxed{ \ y = f(x) \ }[/tex] to the right h units.

    Therefore, the combination of vertical and horizontal shifts is as follows:  

    [tex]\boxed{\boxed{ \ y = f(x \pm h) \pm v \ }}[/tex]  

    The plus or minus sign follows the direction of the shift, i.e., up-down or left-right.

    - - - - - - - - - -

    Let's solve the problem.

    Initially, the graph of [tex]y = \sqrt{x}[/tex] is shifted 2 units up.

    [tex]\boxed{y = \sqrt{x} \rightarrow is \ shifted \ 2 \ units \ up \rightarrow \boxed{ \ y = \sqrt{x} + 2 \ }}[/tex]

    Followed by shifting 5 units left.

    [tex]\boxed{y = \sqrt{x} + 2 \rightarrow is \ shifted \ 5 \ units \ left \rightarrow \boxed{ \ y = \sqrt{x + 5} + 2 \ }}[/tex]

    Thus, the equation that represents the new graph is [tex]\boxed{\boxed{ \ y = \sqrt{x + 5} + 2 \ }}[/tex]

    The answer is A.

    Learn more

    1. Which phrase best describes the translation from the graph y = 2(x – 15)² + 3 to the graph of y = 2(x – 11)² + 3? https://brainly.com/question/1369568
    2. The similar problem of shifting https://brainly.com/question/2488474  
    3. What transformations change the graph of (f)x to the graph of g(x)? https://brainly.com/question/2415963

    Keywords: the graph of, y = √x, shifted 2 units up, 5 units left, which, the equation, represents, the new graph, horizontal, vertical, transformation geometry, translation

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