Graph the equation.
$y=3|x+5|+3$

Identify the vertex of the absolute value function: ( − 5 , 3 ) .
Find the y-intercept by setting x = 0 : ( 0 , 18 ) .
Find additional points, such as when x = − 4 and x = − 6 : ( − 4 , 6 ) and ( − 6 , 6 ) .
Plot the vertex and the points to draw the V-shaped graph: y = 3∣ x + 5∣ + 3 ​ .

Explanation

Understanding the Function We want to graph the absolute value function y = 3∣ x + 5∣ + 3 . This function is a transformation of the basic absolute value function y = ∣ x ∣ . Let's break down the transformations to understand how the graph will look.

Horizontal Shift The parent function is y = ∣ x ∣ , which has a V-shape with its vertex at the origin (0,0). The term ∣ x + 5∣ represents a horizontal shift. Specifically, it shifts the graph of y = ∣ x ∣ to the left by 5 units. So, the vertex of y = ∣ x + 5∣ is at (-5,0).

Vertical Stretch The term 3∣ x + 5∣ represents a vertical stretch by a factor of 3. This makes the V-shape narrower, but the vertex remains at (-5,0).

Vertical Shift Finally, the term 3∣ x + 5∣ + 3 represents a vertical shift upward by 3 units. This moves the vertex from (-5,0) to (-5,3). So, the vertex of the given function y = 3∣ x + 5∣ + 3 is at (-5,3).

Finding the y-intercept To graph the function, we need to find a few more points. Let's find the y-intercept by setting x = 0 : y = 3∣0 + 5∣ + 3 = 3∣5∣ + 3 = 3 ( 5 ) + 3 = 15 + 3 = 18 So, the y-intercept is (0,18).

Finding Additional Points Now, let's find two more points to help us draw the graph. We can choose x values on either side of the vertex x = − 5 . Let's choose x = − 4 and x = − 6 :
For x = − 4 :
y = 3∣ − 4 + 5∣ + 3 = 3∣1∣ + 3 = 3 ( 1 ) + 3 = 6 So, the point is (-4,6). For x = − 6 :
y = 3∣ − 6 + 5∣ + 3 = 3∣ − 1∣ + 3 = 3 ( 1 ) + 3 = 6 So, the point is (-6,6).

Plotting the Graph Now we have the vertex (-5,3), the y-intercept (0,18), and two additional points (-4,6) and (-6,6). We can plot these points and draw the V-shaped graph of the absolute value function.

Final Answer The graph of y = 3∣ x + 5∣ + 3 is a V-shaped graph with vertex at (-5,3), passing through the points (0,18), (-4,6), and (-6,6).


Examples
Absolute value functions are used in many real-world applications, such as measuring distances or deviations from a target value. For example, in manufacturing, absolute value functions can be used to model the tolerance of a machine part. The function y = a ∣ x − b ∣ + c can represent the acceptable range of a measurement x around a target value b , where a determines the sensitivity to deviations, and c represents a baseline or minimum acceptable value. Graphing this function helps visualize the acceptable range and understand how deviations affect the outcome.

Answered by GinnyAnswer | 2025-07-07