https://file.aiquickdraw.com/mathbot/file/1726064255961r7inlvxx.jpg

The image contains four functions for which you are asked to draw the graphs. They are:

1. \( f(x) = -x^2 + 2x - 1 \)
2. \( y = |x| + x \)
3. \( |x| = |y| \)
4. \( y^2 = x \)

Let’s briefly discuss each of these functions and how their graphs look:

1. **\( f(x) = -x^2 + 2x - 1 \)**:  
   This is a quadratic function in the form \( ax^2 + bx + c \), with \( a = -1 \), \( b = 2 \), and \( c = -1 \). It represents a downward-opening parabola. We can find the vertex using the formula \( x = -\frac{b}{2a} \) and then plot the points accordingly.

2. **\( y = |x| + x \)**:  
   This function involves the absolute value of \( x \), which behaves differently for positive and negative values of \( x \). For \( x \geq 0 \), \( y = 2x \), and for \( x < 0 \), \( y = 0 \).

3. **\( |x| = |y| \)**:  
   This represents a symmetrical graph with respect to both the x-axis

Draw graphs of the following functions: f(x) = -x^2 + 2x - 1, y = |x| + x, |x| = |y|, y^2 = x

Explore graphing techniques for a quadratic function, an absolute value function, and symmetric graphs involving absolute values and parabolas. This problem is ideal for high school students in Grades 9-12 and covers key concepts like parabolas, symmetry, and vertex calculations.

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