Exercises 27-36 tell how many units and in what directions...

Exercises $27-36$ tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ x^{2}+y^{2}=25 \quad \text { Up } 3, \text { left } 4 $$

Exercises $27-36$ tell how many units and in what directions the graphs
of the given equations are to be shifted. Give an equation for the
shifted graph. Then sketch the original and shifted graphs together,
labeling each graph with its equation.
$$
x^{2}+y^{2}=25 \quad \text { Up } 3, \text { left } 4
$$

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Key Concepts

Equation Transformation

Transforming the equation of a graph to reflect a translation involves substituting the original variables with expressions that incorporate the translation distances. This process yields a new equation that represents the same geometric shape as the original, but located at a different position on the coordinate plane.

Vertical Translation

A vertical translation involves moving a graph up or down along the y-axis. This can be achieved by adding or subtracting a constant to the entire function, where adding moves the graph upward and subtracting moves it downward.

Translation of Graphs

Translation in coordinate geometry involves moving a graph from one location to another without altering its shape, size, or orientation. Every point of the graph is shifted by the same distance in a specific direction, resulting in congruent figures in different positions on the coordinate plane.

Horizontal Translation

A horizontal translation shifts a graph left or right along the x-axis. This is accomplished by replacing the x-variable with an expression (x - h), where h represents the shift magnitude; a positive h moves the graph right, while a negative h moves it left.

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