Problem 42 Graph. $$y=25-|x|$$... [FREE SOLUTION] (2024)

Chapter 0: Problem 42

Graph. $$y=25-|x|$$

Short Answer

Expert verified

Plot the vertex at (0, 25) and draw lines to form a V-shape using y = 25 - x for x >= 0 and y = 25 + x for x < 0.

Step by step solution

Understanding the Equation

The given equation is \[ y = 25 - |x| \]. This is a function that describes a graph. The absolute value expression $ |x| $ determines the shape of the graph.

Identify Key Points

Determine the key points of the graph. The vertex is at the point where $ x = 0 $. At this point, \[ y = 25 - |0| = 25 \]. So, the vertex is at \[ (0, 25) \].

Evaluate the Function for Other Values

Calculate the function for other values of $ x $ to understand the linear parts of the graph. For example, if $ x = 5 $, then \[ y = 25 - |5| = 20 \]. Similarly, if $ x = -5 $, \[ y = 25 - |-5| = 20 \].

Determine Symmetry

Since $ |x| $ indicates absolute value, the graph is symmetric with respect to the $ y $-axis. This means that for every point \[ (a, b) \], there is a corresponding point \[ (-a, b) \].

Draw Two Straight Lines

The equation can be interpreted as two linear equations split at $ x = 0 $. For $ x \geq 0 $, the line is \[ y = 25 - x \] and for $ x < 0 $, the line is \[ y = 25 + x \]. Draw these lines starting from the vertex. This produces a V-shaped graph.

Graph the Function

Plot the vertex at \[ (0, 25) \], then draw the two lines as described in Step 5. Combine these to reflect the graph of \[ y = 25 - |x| \]. Ensure symmetry across the y-axis.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex form

The vertex form of a function is essential for understanding its graph. In this problem, the given function \[ y = 25 - |x| \]has its vertex clearly indicated. The vertex form helps us find this crucial point, because at $ x = 0 $,all absolute value transformations hit their highest or lowest point. For our equation, when $ x = 0 $,we have:\[ y = 25 - |0| = 25 \]Thus, the vertex is at $ (0, 25) $. This point is where the graph changes direction and starts to form its characteristic 'V' shape.

symmetry

Graphing functions with absolute values typically involves symmetry. Absolute value expressions,like $ |x| $,mean every positive $ x $has a corresponding negative value, both producing the same $ y $-value. This results in a graph that is symmetric about the $ y $-axis. For example, in our function $ y = 25 - |x| $,if we calculate $ y $for $ x = 5 $and $ x = -5 $, we get:\[ y = 25 - |5| = 20 \]and\[ y = 25 - |-5| = 20 \] This symmetry consistently shows up across all $ x $ values, giving a mirrored appearance around the $ y $-axis.

linear equations

Breaking down the absolute value function into linear equations makes graphing straightforward. An absolute value function like $ y = 25 - |x| $splits into two linear parts around $ x = 0 $:

For $ x \geq 0 $: $ y = 25 - x $
For $ x < 0 $: $ y = 25 + x $

The slopes of these two linear parts are different but consistent. The $ y $-intercept remains $ 25 $,and from there—whether moving right or left—the graph slopes down at the same rate of 1 unit per 1 unit of $ x $. This method ensures the structure forms a 'V' shape as expected.

key points

Key points help us sketch the graph precisely. Besides the vertex $ (0, 25) $,identifying other critical points aids in defining the lines. For instance, knowing the function's result at $ x = \pm 5 $provides the point $ (5, 20) $and $ (-5, 20) $.
To do this:

Choose several positive values of $ x $
Compute the corresponding $ y $values
Repeat for negative $ x $values

With points like $ (3, 22) $and $ (-3, 22) $,we gain a clearer picture of the graph formation. Plot these points, connect them with the vertex, and we accurately represent the absolute value function's graph.

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