Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x)        ƒ₂(x)
 x²          |x|²

Verified step by step guidance

1. Start by graphing the function ƒ₁(x) = x². This is a standard parabola that opens upwards with its vertex at the origin (0,0). The graph is symmetric about the y-axis.
2. Next, graph the function ƒ₂(x) = |x|². Note that |x| is the absolute value of x, which means it is always non-negative. However, since |x|² = x² for all real numbers x, the graph of ƒ₂(x) will be identical to the graph of ƒ₁(x).
3. Observe that applying the absolute value function to x before squaring it does not change the graph of the function. This is because squaring a number, whether positive or negative, results in a positive value, which is the same as squaring the absolute value of the number.
4. Consider the implications of the absolute value function. While |x| affects the graph by ensuring all x-values are non-negative, squaring negates this effect, resulting in the same graph as x².
5. Conclude that the application of the absolute value function in ƒ₂(x) does not affect the graph of ƒ₁(x) in this case, as both functions result in the same parabola y = x².

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x-values) and output (y-values) of a function. For the functions ƒ₁(x) = x² and ƒ₂(x) = |x|², understanding their shapes is crucial. ƒ₁ is a standard parabola opening upwards, while ƒ₂, due to the absolute value, also forms a parabola but reflects any negative x-values to positive, ensuring all outputs are non-negative.

Absolute Value Function

The absolute value function, denoted as |x|, transforms any negative input into its positive counterpart. This means that for the function ƒ₂(x) = |x|², the output is always non-negative, regardless of whether x is positive or negative. This property affects the graph by ensuring that all points lie above the x-axis, creating a symmetric shape about the y-axis.

Transformation of Graphs

Transformations of graphs refer to changes made to the original function that affect its position or shape. In this case, applying the absolute value to ƒ₁(x) = x² to create ƒ₂(x) = |x|² results in a graph that retains the parabolic shape of ƒ₁ but modifies it to reflect any negative portions of the graph upwards. This transformation emphasizes the non-negativity of the output values, altering the visual representation of the function.