Question

Graph the system of quadratic inequalities. (See Example 3.) \(y \geq 2 x^2+x-5\) \(y<-x^2+5 x+10\)

Short answer

Graph the parabola for each inequality where \(y = 2x^2 + x - 5\) should include all area above, and \(y = -x^2 + 5x + 10\) should include all area below. The common region satisfying both conditions is the solution to the system of inequalities.

Step-by-step solution

Graph the first inequality

Start by representing the quadratic equation \(y = 2x^2 + x - 5\), which is a parabola opening upwards. As the inequality is greater than or equal to, the area above the parabola will represent the possible y-values for each x within the inequality system.

Graph the second inequality

Next, draw the quadratic equation \(y = -x^2 + 5x + 10\), a parabola opening downwards. As the inequality is less than, the area below this parabola represents the possible y-values for each x according to the second inequality.

Identify the solution region

The solution to the system of inequalities will be identified by the region that fits both criteria, i.e., lies above the first graph and below the second graph. This common region between the two inequalities is the solution.

Key concepts

Quadratic Functions

Quadratic functions are an essential part of algebra that describe a specific form of polynomial equation, typically expressed as \(y = ax^2 + bx + c\). The letter \(a\) dictates the parabola's direction:

• If \(a > 0\), the parabola opens upwards, making it look like a "U".
• If \(a < 0\), the parabola opens downwards, resembling an upside-down "U".

These functions create smooth, symmetrical curves called parabolas. The vertex of the parabola is a crucial component, representing the minimum or maximum point of the function.
Quadratic functions have a line of symmetry that can be found using \(x = -\frac{b}{2a}\). This line makes it easier to predict the shape and position of the graph. Understanding these basics helps when graphing or solving quadratic inequalities.

Systems of Inequalities

A system of inequalities involves two or more inequations that must be satisfied at the same time. In the context of quadratic functions, these inequalities can either be linear or quadratic. Solving a system means finding all possible values that satisfy each inequality simultaneously, which means looking for the common area or region where all conditions are true.
To solve a system of inequalities:

• Graph each inequality separately.
• Note the shaded area that each graph represents.
• The solution is the intersection of these shaded regions, meaning the area where all shades overlap.

This shared region often forms a complex shape, influenced by the nature of each quadratic equation in the system. It's important to remember that solutions to systems may not always fit within neat boundaries, especially with curves involved.

Graphing Parabolas

Graphing parabolas involves several steps to ensure accuracy. Each parabola reflects the equation's unique components, particularly the sign and size of \(a\), \(b\), and \(c\) in the quadratic equation \(y = ax^2 + bx + c\).
Here are practical steps to graph a parabola:

• Determine the direction of the parabola by checking the sign of \(a\).
• Find the vertex, the turning point of the parabola. Use \(x = -\frac{b}{2a}\) to calculate the vertex's x-coordinate and substitute it back to find the y-coordinate.
• Identify the y-intercept, which is \(c\) in the equation, where the graph crosses the y-axis.
• Draw the axis of symmetry through the vertex to guide your graph.
• Plot a few points on either side of the vertex to define the curve well.

Graphing inequalities in relation to parabolas involves shading the areas either above or below the parabola, based on the inequality's direction. This visual approach helps in clearly identifying solutions or overlapping regions within quadratic systems.