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What is the effect on the graph of the function
f(x)=x^(2) when
f(x) is changed to
f(x)+ 9?

What is the effect on the graph of the function $f(x)=x^{2}$ when $f(x)$ is changed to $f(x)+$ $9$ ?

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Find the vertex of the transformed function

Full solution

Q. What is the effect on the graph of the function

f(x)=x^{2}

when

f(x)

is changed to

f(x)+

9

?

Original Function Analysis: We have the original function: $\newline$ $f(x) = x^2$ $\newline$ We need to determine the effect on the graph when the function is changed to: $\newline$ $f(x) + 9$ $\newline$ This results in a new function: $\newline$ $g(x) = f(x) + 9 = x^2 + 9$
Effect of Adding $9$ : The transformation from $f(x)$ to $g(x)$ involves adding $9$ to the original function. In terms of graph transformations, adding a constant to the function value ( $y$ -value) results in a vertical shift of the graph.
Vertical Shift Explanation: Since we are adding $9$ to the entire function, the graph of $g(x)$ will be the graph of $f(x)$ shifted $9$ units upwards. This means that every point $(x, y)$ on the graph of $f(x)$ will be moved to $(x, y + 9)$ on the graph of $g(x)$ .
Visualization of Vertex Shift: To visualize this, consider the vertex of the parabola $y = x^2$ , which is at $(0, 0)$ . When we add $9$ to the function, the new vertex will be at $(0, 0 + 9)$ , which is $(0, 9)$ .
Overall Graph Transformation: The effect on the graph of the function $f(x) = x^2$ when changed to $f(x) + 9$ is a vertical shift upwards by $9$ units. The shape of the graph remains the same, but every point on the graph is now $9$ units higher than it was before.

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