Graph the equation.

[tex]y=-4|x+5|[/tex]

The equation is an absolute value function with vertex at ( − 5 , 0 ) .
The graph opens downwards due to the negative sign.
The y-intercept is at ( 0 , − 20 ) .
The graph is a V-shape with the vertex at ( − 5 , 0 ) , so the final answer is the graph of y = − 4∣ x + 5∣ .

Explanation

Understanding the Equation We are asked to graph the equation y = − 4∣ x + 5∣ . This is an absolute value function, which means it will have a V-shape. The absolute value function ∣ x ∣ has its vertex at x = 0 . The function ∣ x + 5∣ has its vertex at x = − 5 , because that's where the expression inside the absolute value becomes zero. The negative sign in front of the 4 reflects the graph across the x-axis, so instead of opening upwards, it opens downwards. The 4 stretches the graph vertically by a factor of 4.

Finding the Vertex The vertex of the graph is the point where the absolute value expression is zero. This occurs when x + 5 = 0 , which means x = − 5 . When x = − 5 , we have y = − 4∣ ( − 5 ) + 5∣ = − 4∣0∣ = 0 . So the vertex is at the point ( − 5 , 0 ) .

Finding the x-intercept The x-intercept is the point where the graph crosses the x-axis, which means y = 0 . So we set y = 0 and solve for x : 0 = − 4∣ x + 5∣ Dividing both sides by -4, we get: 0 = ∣ x + 5∣ This means x + 5 = 0 , so x = − 5 . The x-intercept is at ( − 5 , 0 ) , which is also the vertex.

Finding the y-intercept The y-intercept is the point where the graph crosses the y-axis, which means x = 0 . So we set x = 0 and solve for y : y = − 4∣0 + 5∣ = − 4∣5∣ = − 4 ( 5 ) = − 20 So the y-intercept is at ( 0 , − 20 ) .

Finding Additional Points To get a better idea of the shape of the graph, let's find a couple more points. Let's try x = − 6 : y = − 4∣ ( − 6 ) + 5∣ = − 4∣ − 1∣ = − 4 ( 1 ) = − 4 So the point ( − 6 , − 4 ) is on the graph. Now let's try x = − 4 : y = − 4∣ ( − 4 ) + 5∣ = − 4∣1∣ = − 4 ( 1 ) = − 4 So the point ( − 4 , − 4 ) is on the graph.

Graphing the Equation Now we can plot the points we found: the vertex ( − 5 , 0 ) , the y-intercept ( 0 , − 20 ) , and the points ( − 6 , − 4 ) and ( − 4 , − 4 ) . The graph is a V-shape opening downwards, with the vertex at ( − 5 , 0 ) .


Examples
Absolute value functions are used in many real-world applications, such as calculating distances or tolerances in engineering and manufacturing. For example, if you're designing a part that needs to be within a certain tolerance of a specific size, you can use an absolute value function to model the acceptable range of sizes. Also, absolute value functions are used in computer graphics to create reflections and other special effects. Understanding how to graph and manipulate absolute value functions is a valuable skill in many fields.

Answered by GinnyAnswer | 2025-07-05

The graph of the equation y = − 4∣ x + 5∣ features a V-shape that opens downwards, with its vertex at ( − 5 , 0 ) . The y-intercept is located at ( 0 , − 20 ) , and the graph crosses the x-axis at ( − 5 , 0 ) . Additional points include ( − 6 , − 4 ) and ( − 4 , − 4 ) .
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Answered by Anonymous | 2025-07-19