Problem 18 - The Calculus with Analytic Geometry

Question

In Exercises 7 through 28 , draw a sketch of the graph of the equation. $$ y=-|x|+2 $$

Step-by-Step Solution

Answer

Draw an upside-down V with vertex at (0,2).

Step 1: Identify the Base Function

The base function is the absolute value function, which is written as \( y = |x| \). This is a V-shaped graph that opens upwards and is symmetric about the y-axis.

Step 2: Apply the Negation

The equation given is \( y = -|x| + 2 \). Negating the absolute value function, \( -|x| \), flips the V-shaped graph upside down. Now the graph opens downwards.

Step 3: Apply the Vertical Shift

The equation has a +2 at the end, which means the entire graph is shifted 2 units upwards. This moves the vertex of the upside-down V from (0,0) to (0,2).

Step 4: Sketch the Graph

Combine the information from the previous steps to sketch the graph. Start at the vertex (0,2) and draw two lines extending downwards to the left and right symmetrically. The left part of the graph will follow the line \( y = -x + 2 \) and the right part will follow the line \( y = x + 2 \) when considering the absolute value.

Key Concepts

absolute value function

The absolute value function is a fundamental concept in algebra. Generally written as \( y = |x| \), it produces a V-shaped graph. This graph opens upwards and is symmetric about the y-axis. The absolute value function measures the distance of a number from zero, giving only non-negative outputs.
Understanding the basic shape and properties of \( y = |x| \) is crucial before applying any transformations.

graph transformations

Graph transformations help alter the appearance of the base function. For the given problem, we start with the base function \( y = |x| \). There are many types of transformations:

• Vertical Shifts
• Horizontal Shifts
• Reflections
• Stretching and Compressing

These transformations can change the position, orientation, and shape of the original graph.
For instance, the equation \( y = -|x| + 2 \) involves a reflection and a vertical shift.

vertex of a graph

The vertex of a graph is a key feature, particularly in absolute value functions. For \( y = |x| \), the vertex is the point where the graph changes direction, located at (0,0).
In the given problem, transformations shift the vertex:

• The negation affects the direction (reflection)
• The vertical shift changes the vertical position

Hence, for \( y = -|x| + 2 \), the vertex moves to (0,2).

vertical shifts

Vertical shifts move the graph up or down without changing its shape. This is done by adding or subtracting a constant to the function. In our exercise, the +2 at the end of \( y = -|x| + 2 \) shifts the graph 2 units upwards.
Hence, what was originally at (0,0) for \( y = -|x| \), becomes (0,2) for \( y = -|x| + 2 \).

reflection of functions

Reflections flip the graph over a specific axis. The equation \( y = -|x| \) involves a reflection over the x-axis. This means our V-shaped graph, which originally opens upwards, now opens downwards.
The reflection affects the direction of our graph, making it open downwards but still symmetric about the y-axis.