Absolute Value Graphs Explained with Animations

Welcome to our exploration of the absolute value function!Let's start with the basic definition: f of x equals the absolute value of x.To understand this function, let's look at its graph on a coordinate plane.The absolute value function creates a V-shaped graph. Let's trace it to see how it works.Starting from the left, notice how negative inputs produce positive outputs.As we continue to the right, the output remains positive.This function has several important characteristics. First, it's symmetric about the y-axis.The vertex is at the origin, where x equals zero.And most importantly, the function always outputs non-negative values, meaning the graph never goes below the x-axis.This means all outputs of the absolute value function are either positive or zero.Now that we understand the basic absolute value function, let's explore what happens when we add or subtract constants.When we add a positive number like 2, the entire graph shifts upward by 2 units. Notice how the V-shape maintains its form, but moves up.The vertex of our shifted function is now at the point (0, 2). Every point on the graph has moved up 2 units from the original function.Now, let's subtract 3 from our function. Watch how the entire graph shifts downward by 3 units while maintaining its V-shape.The vertex has now moved to the point (0, -3). Every point on the graph is 3 units below its position in the original function.Let's compare all three functions side by side. Notice how each vertical shift creates a parallel version of the original absolute value function.Remember these key points: Adding a constant shifts the graph up, subtracting shifts it down, and the V-shape is always preserved.Now we'll explore how the absolute value function moves horizontally.When we subtract a number inside the absolute value, like in f of x equals absolute value of x minus 2, the graph shifts 2 units right.Notice how subtracting inside the absolute value causes a right shift.When we add a number inside the absolute value, like in f of x equals absolute value of x plus 3, the graph shifts 3 units left.Adding inside the absolute value results in a left shift.Remember this key rule: Adding inside the absolute value shifts left, while subtracting shifts right.As we move along the shifted graph, notice how each point maintains the same V-shape, just centered at a different location.Now we'll explore how multiplying the absolute value function by a positive number affects its shape.The original absolute value function has slopes of positive and negative one.When we multiply by a number greater than one, like two, the function becomes steeper.Notice how the slopes are now positive and negative two, making the V shape steeper.Making the multiplier even larger, like three, stretches the function even more.Now, let's look at what happens when we multiply by a fraction, like one-half.The slopes are now one-half, making the V shape wider and more gentle.Using an even smaller multiplier, like one-third, compresses the function further.Let's compare different stretches and compressions side by side.Notice how the vertex remains at the origin while the slopes change based on the multiplier.Remember, multipliers greater than one stretch the graph vertically, while multipliers between zero and one compress it.Now we'll combine all our transformations to graph a complex absolute value function.Our function is f of x equals two times the absolute value of x minus three plus one.First, we handle what's inside the parentheses. X minus three means we shift the graph right three units.Next, we apply the absolute value, creating our characteristic V shape at x equals three.Then we multiply by two, which stretches the graph vertically, making it steeper.Finally, we add one, shifting the entire graph up one unit.Our final graph has its vertex at the point (3,1), and passes through the points (1,5) and (5,5).The slopes of the two pieces are negative two and positive two, due to our vertical stretch factor.