While a quadratic *equation* gives you a parabola, a quadratic *inequality* describes an entire region of the coordinate plane. Graphing these inequalities involves finding the boundary parabola and then determining which side of it to shade. Our Graphing Quadratic Inequalities Calculator is a tool designed to help algebra students understand the properties of these graphs. Instead of drawing the graph for you, it provides a detailed description of all its key features—the vertex, the intercepts, the direction, and the shading—which helps you learn what to look for when graphing by hand.
How to Use the Graphing Quadratic Inequalities Calculator
Analyzing the graph of your inequality is a simple process:
- Enter the Coefficients: For an inequality in the form y > ax² + bx + c, input the numerical values for 'a', 'b', and 'c'.
- Select the Operator: Choose the correct inequality symbol from the dropdown menu (<, ≤, >, or ≥).
- Analyze the Graph: Click the "Analyze" button.
- View the Properties: The calculator will display a list of the key properties of the graph, including its vertex, intercepts, and shading direction.
Understanding the Components of the Graph
The solution to a two-variable quadratic inequality is a region on the x-y plane. The graph has two main components: the boundary line (a parabola) and the shaded area.
1. The Boundary Parabola
The first step is to consider the related equation y = ax² + bx + c. This equation defines the parabolic boundary of your solution region. Our calculator provides the key features of this parabola:
- Vertex: This is the turning point of the parabola—either its lowest point (a minimum) or its highest point (a maximum).
- Y-Intercept: This is the point where the parabola crosses the vertical y-axis. It's found by setting x=0 in the equation.
- X-Intercepts (Roots): These are the points where the parabola crosses the horizontal x-axis. A parabola can have two, one, or no real x-intercepts.
- Direction: The parabola "opens up" if the 'a' coefficient is positive, and "opens down" if 'a' is negative.
2. The Boundary Line Style: Solid vs. Dashed
The style of the parabolic line indicates whether the points *on* the parabola are part of the solution.
- A solid line is used for inequalities with "or equal to" (≤ or ≥). This means that points on the parabola itself are included in the solution set.
- A dashed line is used for strict inequalities (< or >). This signifies that points on the parabola are *not* part of the solution; they are only the boundary.
3. The Shaded Region
The shading indicates which region of the coordinate plane contains all the (x, y) points that make the inequality true.
- For "greater than" inequalities (y > ... or y ≥ ...), you shade the region above the parabola.
- For "less than" inequalities (y < ... or y ≤ ...), you shade the region below the parabola.
A simple way to confirm this is to pick a test point, like the origin (0,0), and plug it into the inequality. If it results in a true statement, you shade the region containing that point. If it's false, you shade the other region.
Frequently Asked Questions
How do I find the vertex of a parabola?
For a quadratic in standard form (ax² + bx + c), the x-coordinate of the vertex can be found with the formula x = -b / 2a. Once you have the x-coordinate, you can plug it back into the equation to solve for the corresponding y-coordinate.
What's the difference between this and graphing a single-variable inequality on a number line?
A single-variable inequality (e.g., x² - 4 > 0) has solutions that are ranges of x-values, which are graphed on a one-dimensional number line. A two-variable inequality (e.g., y > x² - 4) has solutions that are pairs of (x, y) coordinates, which represent an entire region on the two-dimensional coordinate plane.
How can I be sure which side of the parabola to shade?
The "test point" method is foolproof. Choose a simple point that is clearly not on the parabola, like the origin (0,0) if possible. Substitute its x and y coordinates into your original inequality. If the resulting statement is true (e.g., 0 > -3), you shade the entire region containing that test point. If the statement is false (e.g., 0 > 5), you shade the region on the other side of the parabola.