Problem 22 - Precalculus Essentials

Question

Graph each equation.Let \(x=-3,-2,-1,0\) \(1,2,\) and 3 $$y=-2|x|$$

Step-by-Step Solution

Answer

By substituting the given x-values into the equation, calculating the corresponding y-values, and sketching these points you can represent \(y=-2|x|\) which is an inverted 'V' shape graph.

Step 1: Write down the equation

The given equation is \(y=-2|x|\). For each x-value in \(-3, -2, -1, 0, 1, 2, 3\), substitute it into the equation to find the corresponding y-value.

Step 2: Substitute and calculate

Start substituting x-values. For x=-3, \(y=-2|-3| = -2*3 = -6\). Continue this until you have calculated the y-value for each x-value.

Step 3: Plot the points

Now that each x-value has a corresponding y-value, plot these points on the graph. You should notice that for positive x, the graph reflects onto the negative side, resulting in a 'V' shape. This is due to the absolute value operation, and the negative sign in the equation makes the graph inverted.

Step 4: Sketch the graph

Finally, connect all the plotted points with a straight line, creating an inverted 'V' shape. This is the graph of the equation \(y=-2|x|\).

Key Concepts

Graph Transformations

When we talk about graph transformations, we are discussing how a graph changes in position or shape due to alterations in its equation. In this exercise, the equation given was \( y = -2|x| \). To grasp the transformations here, let's consider what each part of the equation does.

• The absolute value \(|x|\) produces a V-shaped graph due to its definition as a distance from zero.
• When multiplied by -2, the graph flips vertically. The "-" sign causes the graph to invert, making it open downwards.
• The "2" stretches the graph away from the x-axis, enlarging the V-shape.

The transformation affects the graph's orientation and size. If you look at the original absolute value graph, \( y = |x| \), it opens upwards. Meanwhile, \( y = -2|x| \) opens downwards and is wider. Understanding these transformations helps in swiftly sketching and analyzing different functions.

Piecewise Functions

Piecewise functions consist of different expressions based on different intervals of the input value. While not explicitly stated in this exercise, the absolute value function can be thought of as a piecewise function:

• For \( x \geq 0 \), \( |x| = x \).
• For \( x < 0 \), \( |x| = -x \).

This duality is why absolute value graphs appear V-shaped. It behaves like two separate linear functions joined at a single point; the vertex. In our transformed graph, \( y = -2|x| \), the function can also be seen as:

• \( y = -2x \) when \( x \geq 0 \).
• \( y = 2x \) when \( x < 0 \).

These segments create a smooth transition at the vertex to form the inverted V-shape. Breaking down graphs into piecewise segments like this can make them easier to analyze and sketch.

Precalculus

Precalculus serves as the mathematical foundation preparing students for calculus. It often involves manipulating functions, understanding their graphs, and learning algebraic expressions. This exercise is a good example of honing skills in these areas:

• Function Analysis: Understanding the effect of absolute value and transformations on function behavior is key.
• Graphing Skills: The task strengthens the ability to graph equations through plotted points and recognition of shapes, here an inverted V.
• Critical Thinking: Recognizing transformations without needing to compute each step encourages abstract thinking.

By mastering these aspects, students build a strong analytical foundation, which becomes increasingly important as they advance into calculus and higher-level mathematics. Familiarity with transformations and function behavior gained in precalculus ensures students are well-prepared for future challenges in math.