Graph the equation.
$y=4|x-3|+4$

Identify the equation as an absolute value function y = 4∣ x − 3∣ + 4 with vertex at ( 3 , 4 ) .
Calculate the y-intercept by setting x = 0 , resulting in y = 16 .
Determine that there are no x-intercepts since ∣ x − 3∣ = − 1 has no solution.
Plot the vertex, y-intercept, and additional points to graph the V-shaped function: y = 4∣ x − 3∣ + 4 ​ .

Explanation

Analyze the equation We are asked to graph the equation y = 4∣ x − 3∣ + 4 . This is an absolute value function, which means it will have a V-shape. 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 this case, a = 4 , h = 3 , and k = 4 .

Find the vertex The vertex of the absolute value function is at ( h , k ) = ( 3 , 4 ) . This is the point where the V-shape changes direction.

Find the y-intercept To find the y-intercept, we set x = 0 and solve for y : y = 4∣0 − 3∣ + 4 = 4∣ − 3∣ + 4 = 4 ( 3 ) + 4 = 12 + 4 = 16 So the y-intercept is at ( 0 , 16 ) .

Find the x-intercepts To find the x-intercept, we set y = 0 and solve for x : 0 = 4∣ x − 3∣ + 4 − 4 = 4∣ x − 3∣ − 1 = ∣ x − 3∣ Since the absolute value of any number is non-negative, there is no solution for x . This means there are no x-intercepts.

Find additional points Now, let's find a few additional points to help us graph the function. We can choose some values for x that are close to the vertex at x = 3 .


If x = 1 , then y = 4∣1 − 3∣ + 4 = 4∣ − 2∣ + 4 = 4 ( 2 ) + 4 = 8 + 4 = 12 . So the point is ( 1 , 12 ) .
If x = 2 , then y = 4∣2 − 3∣ + 4 = 4∣ − 1∣ + 4 = 4 ( 1 ) + 4 = 4 + 4 = 8 . So the point is ( 2 , 8 ) .
If x = 4 , then y = 4∣4 − 3∣ + 4 = 4∣1∣ + 4 = 4 ( 1 ) + 4 = 4 + 4 = 8 . So the point is ( 4 , 8 ) .
If x = 5 , then y = 4∣5 − 3∣ + 4 = 4∣2∣ + 4 = 4 ( 2 ) + 4 = 8 + 4 = 12 . So the point is ( 5 , 12 ) .

Plot the points and draw the graph We have the vertex at ( 3 , 4 ) , the y-intercept at ( 0 , 16 ) , and additional points at ( 1 , 12 ) , ( 2 , 8 ) , ( 4 , 8 ) , and ( 5 , 12 ) . We can plot these points and draw the V-shaped graph. The graph opens upwards since the coefficient of the absolute value term is positive ( a = 4 ).

Final Answer The graph of the equation y = 4∣ x − 3∣ + 4 is a V-shaped graph with the vertex at ( 3 , 4 ) , y-intercept at ( 0 , 16 ) , and no x-intercepts.


Examples
Absolute value functions are used in many real-world applications, such as measuring distances, modeling tolerances in manufacturing, and representing error bounds in scientific measurements. For example, in manufacturing, the acceptable deviation from a target dimension can be modeled using an absolute value function. If a machine part is supposed to be 3 cm long, a tolerance of 0.1 cm can be expressed as $|x-3|
< 0.1$, where x is the actual length of the part. This ensures that the part is within acceptable limits. Similarly, in navigation, the distance from a planned route can be modeled using absolute value to ensure the vehicle stays on course. Understanding absolute value functions helps in controlling and analyzing deviations from desired values in various practical scenarios.

Answered by GinnyAnswer | 2025-07-07