The absolute value of a number is its distance from zero, meaning the result is always non-negative. Graphing an absolute value equation results in a distinctive V-shape. Understanding how to translate the algebraic expression into this visual representation is a fundamental skill in graphing functions. This guide details the process for accurately plotting these equations.
The Anatomy of the Absolute Value Graph
The standard, or vertex, form for graphing absolute value equations is $y = a|x – h| + k$. This format is useful because the variables $a$, $h$, and $k$ directly correspond to transformations on the graph. The V-shape originates from the absolute value operation, which forces the function to change direction at a single point.
The parameter $a$ is the coefficient outside the absolute value bars, controlling the vertical stretch or compression and the graph’s direction. Inside the bars, $h$ influences the horizontal shift. The final parameter, $k$, dictates the vertical shift. These three values provide the blueprint for drawing the function.
Step 1: Locating the Vertex
The vertex is the point where the two linear segments meet and the graph changes direction. In the standard form $y = a|x – h| + k$, the coordinates of the vertex are $(h, k)$. The value of $k$ is taken directly from the equation, corresponding to the vertical shift.
The horizontal shift $h$ is always the opposite sign of the value written inside the absolute value expression. For example, in an equation like $y = 2|x – 5| + 1$, the $h$ value is positive 5 because the standard form is defined with subtraction. Conversely, if the equation is $y = |x + 3| – 4$, the expression $x + 3$ must be interpreted as $x – (-3)$, making $h$ negative 3. The vertex for this example is $(-3, -4)$.
Step 2: Determining Direction and Slope
The coefficient $a$ determines the graph’s opening direction and steepness. If $a$ is positive, the graph opens upward, forming the standard “V” shape. If $a$ is negative, the graph is reflected across the x-axis and opens downward, creating an inverted “V.”
The magnitude of $a$ dictates the slope of the two linear segments extending from the vertex. This value can be interpreted as a rate of change, or rise over run, for the right-hand segment. For instance, if $a$ equals 3, the slope of the right segment is 3, meaning one would move up three units and right one unit from the vertex to find the next point. This magnitude determines the vertical stretch; a value of $|a| > 1$ results in a narrower graph, while a fraction where $0 < |a| < 1$ makes the graph appear wider.
Step 3: Plotting and Connecting Points
Use the information derived from $h$, $k$, and $a$ to sketch the complete graph. First, plot the vertex point $(h, k)$ on the coordinate plane. This point serves as the origin for the two rays that constitute the V-shape.
Next, use the slope determined by $a$ to plot a second point on the right side of the vertex. For example, if $a$ is 2, move up 2 and right 1 from the vertex to mark a new point. The graph is symmetrical around the vertical line that passes through the vertex, known as the axis of symmetry, $x = h$. To find the corresponding point on the left side, simply mirror the second point across this vertical line. Connecting the vertex to the two points with straight lines completes the accurate graph.