Enter Graph Values
Choose a graph mode. Then enter the curve, horizontal line, or endpoint data shown by the graph.
Formula Used
The calculator starts from the absolute value graph form:
y = a|x - h| + k
The vertex is (h, k). The value a controls stretch and reflection. A solid curve uses ≤ or ≥. A dashed curve uses < or >.
For a horizontal boundary y = c, solve:
a|x - h| + k relation c
Then subtract k and divide by a. Reverse the inequality sign when a is negative.
For two boundary points L and R:
h = (L + R) / 2 and r = (R - L) / 2
Inside shading gives |x - h| ≤ r. Outside shading gives |x - h| ≥ r. Open endpoints make the sign strict.
How to Use This Calculator
- Select the mode that matches your graph.
- Enter the vertex values h and k when using a V-shaped graph.
- Enter the stretch value a. Use a negative value for a reflected graph.
- Choose solid or dashed boundary behavior.
- Choose shading above, below, inside, or outside.
- Press the calculate button to see the inequality and interval.
- Use the step list to check the sign, center, and radius.
Example Data Table
| Graph information | Input choice | Expected inequality |
|---|---|---|
| Vertex at (2, 1), opens upward, solid curve, shaded above | Curve mode, a = 1, h = 2, k = 1 | y ≥ |x - 2| + 1 |
| Boundary points -3 and 7, shaded between, closed dots | Boundary point mode | |x - 2| ≤ 5 |
| Graph y = 2|x - 1| - 3 is below y = 5 | Horizontal graph line mode | |x - 1| ≤ 4 |
Understanding Graphs and Absolute Value Inequalities
Absolute value inequalities often begin as pictures. A number line may show two dots and a shaded interval. A coordinate graph may show a V-shaped curve and a shaded region. This calculator reads those visual clues. It turns them into algebraic statements with clear steps.
Why the Vertex Matters
The vertex is the turning point of an absolute value graph. It gives the horizontal shift and vertical shift. In the form y = a|x - h| + k, the vertex is (h, k). The value h becomes the center of many one-variable inequalities. The value k moves the graph up or down. When you know the vertex, most of the expression is already known.
Reading Boundary Style
Graph style decides whether equality is included. A solid line or closed dot includes the boundary. That means the inequality uses ≤ or ≥. A dashed line or open dot excludes the boundary. That means the inequality uses < or >. This small visual detail changes the answer. It also changes interval notation.
Inside and Outside Shading
Number line shading tells the relation. Shading between two boundary points means the distance from the center is small enough. That creates |x - h| ≤ r or |x - h| < r. Shading outside the points means the distance is large enough. That creates |x - h| ≥ r or |x - h| > r. The radius r is half the distance between the boundary points.
Using Horizontal Comparisons
A horizontal line can cut a V-shaped graph. The intersection points create an interval on the x-axis. The calculator solves a|x - h| + k relation c. It subtracts k first. Then it divides by a. If a is negative, the relation must reverse. This keeps the inequality true.
Avoiding Common Mistakes
Many errors come from mixing up center and radius. The center is the midpoint of the two x-boundaries. The radius is the distance from the center to either boundary. Another common error is forgetting to reverse the sign after dividing by a negative stretch value. A third error is using a closed interval when the graph has open endpoints.
Checking Your Answer
After forming an inequality, test one value from each interval. Pick the center, a left outside value, and a right outside value. Substitute each value into the absolute expression. The true tests show where the graph should be shaded. This habit catches reversed signs quickly. It also confirms open and closed endpoints. When a point lies on the boundary, compare it with the selected endpoint style before you record the final solution in interval notation.
Practical Learning Value
This tool is useful for algebra homework, graph checks, and test preparation. It gives the final inequality, interval notation, and reasoning. You can compare your hand solution with each step. You can also change one graph detail and see how the algebra changes. That makes graph interpretation faster, cleaner, and easier to remember.
Frequently Asked Questions
What does this calculator convert?
It converts graph clues into absolute value inequalities. You can use vertex data, shading, boundary style, horizontal comparison lines, or number line boundary points.
What does h mean in the formula?
The value h is the x-coordinate of the vertex. It is also the center when converting two boundary points into one absolute value inequality.
What does k mean in the formula?
The value k is the y-coordinate of the vertex. It shifts the V-shaped graph up or down before the inequality is compared with a horizontal value.
When should I use less than?
Use a less than relation when the graph shows values inside a distance, below a boundary, or between two open endpoints.
When should I use greater than?
Use a greater than relation when the graph shows values outside a distance, above a boundary, or beyond two open boundary points.
Why does a negative stretch reverse the sign?
Dividing an inequality by a negative number reverses the relation. This rule keeps the solution set correct after isolating the absolute value expression.
How do solid and dashed curves affect answers?
A solid curve includes equality, so use ≤ or ≥. A dashed curve excludes equality, so use < or >.
How are boundary points converted?
The calculator finds the midpoint for the center. It finds half the distance for the radius. Then it chooses inside or outside shading.
Can the result be no solution?
Yes. For example, an absolute value cannot be less than a negative number. The calculator flags that case in the interval result.
Can the result be all real numbers?
Yes. An absolute value is always at least zero. Some comparisons with negative limits are true for every real x-value.
What if my graph is reflected downward?
Enter a negative value for a. The calculator will build the reflected graph and reverse the relation during one-variable solving when needed.