Graph the equation.

$y=-4|x+5|$

Find the vertex of the absolute value function: ( − 5 , 0 ) .
Determine the y-intercept by setting x = 0 : ( 0 , − 20 ) .
Determine the direction of opening: downwards since the coefficient of the absolute value is negative.
Sketch the V-shaped graph with the vertex at ( − 5 , 0 ) and passing through ( 0 , − 20 ) .

The final graph is a V-shape opening downwards with vertex ( − 5 , 0 ) . y = − 4∣ x + 5∣ ​
Explanation

Understanding the Equation We want to graph the equation y = − 4∣ x + 5∣ . This is an absolute value function, which means it will have a V-shape. The key features to identify are the vertex, intercepts, and the direction in which the graph opens.

Finding the Vertex The general form of an absolute value function is y = a ∣ x − h ∣ + k , where ( h , k ) is the vertex of the V-shape. In our case, y = − 4∣ x + 5∣ , we can rewrite it as y = − 4∣ x − ( − 5 ) ∣ + 0 . Thus, the vertex is at ( − 5 , 0 ) .

Finding the x-intercept To find the x-intercept, we set y = 0 and solve for x :


0 = − 4∣ x + 5∣
∣ x + 5∣ = 0
x + 5 = 0
x = − 5
So, the x-intercept is ( − 5 , 0 ) , which is also the vertex.

Finding the y-intercept To find the y-intercept, we set x = 0 and solve for y :

y = − 4∣0 + 5∣
y = − 4∣5∣
y = − 4 ( 5 )
y = − 20
So, the y-intercept is ( 0 , − 20 ) .

Determining the Direction and Symmetry Since the coefficient of the absolute value term is − 4 , which is negative, the graph opens downwards. The absolute value function is symmetric about the vertical line passing through the vertex, which is x = − 5 .

Finding Additional Points To plot the graph, we plot the vertex ( − 5 , 0 ) and the y-intercept ( 0 , − 20 ) . We can also find another point to help us sketch the graph. Let's choose x = − 10 :


y = − 4∣ − 10 + 5∣
y = − 4∣ − 5∣
y = − 4 ( 5 )
y = − 20
So, the point ( − 10 , − 20 ) is also on the graph.

Sketching the Graph Now we can sketch the graph. It's a V-shape opening downwards with the vertex at ( − 5 , 0 ) and passing through the points ( 0 , − 20 ) and ( − 10 , − 20 ) .

Examples
Absolute value functions are used in various real-world scenarios, such as modeling distances or errors. For example, in engineering, when designing a system, the absolute value function can be used to model the tolerance or acceptable deviation from a target value. If the target value is -5 and the acceptable deviation is represented by the absolute value of x+5, then the function y = -4|x+5| can represent the cost or penalty associated with deviations from the target, where the cost increases linearly with the deviation.

Answered by GinnyAnswer | 2025-07-05