Question

Graph the system of quadratic inequalities. (See Example 3.) \(y \geq x^2-4\) \(y \leq-2 x^2+7 x+4\)

Short answer

The solution to the system of inequalities is the region which is shaded on both graphs, specifically it's below the parabola \(y= -2x^2+7x+4\) and above the parabola \(y=x^2-4\).

Step-by-step solution

Graph first inequality

First, graph the quadratic inequality \(y \geq x^2-4\). This represents a parabola that opens upwards with vertex at (0,-4). Because of the \(\geq\) inequality, both the line of the parabola and the area above it are included in the solution.

Graph second inequality

Next, graph the quadratic inequality \(y \leq -2x^2+7x+4\). This represents a parabola that opens downwards. To find the vertex, complete square for \(-2x^2+7x\), which gives \(-2(x-7/4)^2+31/8\). So it has vertex at \((7/4, 31/8)\) and intersect y-axis at (0,4). Because of the \(\leq\) inequality, both the line of the parabola and the area below it are included in the solution.

Identify the solution region

The solution to the system of inequalities will be the region where the solutions to both inequalities overlap. This is the region which is shaded on both graphs.

Key concepts

Graphing quadratic functions

Quadratic functions often take the form of \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. They are very important in mathematics and often appear in various applications. To graph a quadratic function:

• Identify the Vertex: The vertex is the highest or lowest point on the graph of a parabola. For a function in standard form \(y = ax^2 + bx + c\), you can find the vertex using the formula \(x = -\frac{b}{2a}\). Substitute this value back into the equation to find the corresponding \(y\)-coordinate.
• Determine the Direction of the Parabola: If \(a > 0\), the parabola opens upwards like a U; if \(a < 0\), it opens downwards like an inverted U.
• Plot Key Points: Identify where the parabola crosses the y-axis, known as the y-intercept, found by setting \(x = 0\). From the vertex, you can plot additional points by selecting \(x\)-values and calculating the corresponding \(y\)-values.
• Draw the Line and Shade Appropriately: If dealing with inequalities, use a dashed line for \(>\) or \(<\), and a solid line for \(\geq\) or \(\leq\). Shade the solution area according to the inequality.

Using these steps properly will lead to the accurate graphing of quadratic functions, allowing you to visualize and solve inequalities effectively.

Parabolas

A parabola is the graph of a quadratic function. Understanding its properties can help in solving quadratic equations and inequalities. Here are some key points:

• Shape and Symmetry: Parabolas are symmetric about their vertical axis, meaning they are mirror images on either side of this line. This axis passes through the vertex.
• Vertex Form and Standard Form: A parabola can be expressed in vertex form \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex. This form is useful for identifying the vertex easily. The standard form is \(y = ax^2 + bx + c\). Transitioning between these forms may require completing the square.
• Intercepts: These include the y-intercept where the parabola crosses the y-axis (at \(x = 0\)), and any x-intercepts where the parabola crosses the x-axis, determined by solving the equation \(ax^2 + bx + c = 0\).
• Direction of Opening: As mentioned, if the coefficient of \(x^2\) (\(a\)) is positive or negative affects whether the parabola opens upwards or downwards.

Recognizing these features of parabolas can greatly aid in your understanding of quadratic graphs and solving related inequalities.

Systems of inequalities

Systems of inequalities involve finding values that satisfy multiple inequalities at once. To solve systems:

• Graph Each Inequality: Start by graphing each inequality on the same coordinate plane. Use shading to represent the solution set of each inequality.
• Check For Overlapping Regions: The solution to the system is where the shaded regions overlap. This overlapping area contains the values that satisfy all the given inequalities.
• Use Test Points: Pick a test point in the overlapped region to confirm it satisfies all inequalities. A point does not have to lie on a curve to be part of the solution unless specified by equality.
• Verify and Interpret Results: Translate the graphical solution back into values or conditions as required by your problem; sometimes numeric estimation is necessary.

Understanding these steps will help you systematically work through systems of inequalities, ensuring no solutions are overlooked and all requirements are met efficiently.