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A parabola is a U-shaped curve that you commonly see in the graph of a quadratic function. Quadratic functions, like the one in the exercise, often take the form \( y = ax^2 + bx + c \) or similar, like our function \( f(x) = 2(x-1)^2 \). The term \( (x-1)^2 \) indicates that the parabola has been shifted horizontally. In this function, the axis of symmetry (a vertical line where the parabola mirrors itself) is at \( x = 1 \). This is derived from the \((x-1)\) part. The number 2 in front of \((x-1)^2\) is what we call the coefficient, and it affects the "width" and "direction" of the parabola. A larger number makes the parabola narrower. Key points about parabolas:- **Vertex**: The highest or lowest point. Here, it is where \( x = 1 \), because the squared term becomes zero.- **Axis of Symmetry**: The vertical line passing through the vertex. In our example, it's \( x = 1 \).- **Direction**: If the coefficient is positive, it opens upward. If negative, downward. In our case, the parabola opens upward.

Evaluating a function means finding the value of the function for a particular \( x \). You often come across problems in which you're given a function and need to substitute a specific \( x \) value to find the corresponding \( f(x) \).In our exercise, the function to evaluate is \( f(x) = 2(x-1)^2 \). We need to compute the outputs for different \( x \) values:- Begin by substituting the \( x \) value into the function.- Compute the result within the parentheses.- Square the result.- Multiply the squared value by the coefficient outside.For instance, when evaluating \( f(-1) \):1. Substitute \( -1 \) into the function, yielding \( 2((-1)-1)^2 \).2. Simplify inside the parentheses to get \( (-2)^2 \).3. Square to get \( 4 \).4. Multiply by \( 2 \) to get \( 8 \).Therefore, \( f(-1) = 8 \). Repeat similar steps for other \( x \) values.

The substitution method is crucial when dealing with functions in algebra. This involves replacing a variable (in our case, \( x \) values) with given numbers to find the corresponding function values.To use substitution effectively:- Identify the function and the numbers you will substitute in.- Carefully replace the \( x \) variable with each number.- Ensure each arithmetic operation is conducted correctly.For example, from the exercise when \( x = 2 \), substitute \( 2 \) into \( f(x) = 2(x-1)^2 \):- You get \( 2(2-1)^2 \), a very straightforward replacement.- Resolve within the parentheses \( 2(1)^2 \).- Calculate the squared result \( 1^2 = 1 \).- Compute the final multiplication \( 2 \times 1 = 2 \).Thus, \( f(2) = 2 \).The substitution method allows for accurately calculating the table of values, paving the way for graphing or further analyses.