Problem 31 - Algebra 2

Question

Graph each inequality. $$ y>-x^{2}+10 x-23 $$

Step-by-Step Solution

Answer

Graph the parabola \( y = -x^{2} + 10x - 23 \) with a dashed line, shade the region above it.

Step 1: Identify the Standard Form

The given inequality is \( y > -x^{2} + 10x - 23 \). This is a quadratic inequality in the form of \( y > ax^{2} + bx + c \), where \( a = -1 \), \( b = 10 \), and \( c = -23 \).

Step 2: Convert to Equation

To graph the boundary of the region defined by the inequality, convert it into an equation: \( y = -x^{2} + 10x - 23 \). This represents a parabola.

Step 3: Determine Parabola Orientation

Since the coefficient of \( x^2 \) (i.e., \( a = -1 \)) is negative, the parabola opens downward.

Step 4: Find the Vertex of the Parabola

The vertex formula for a parabola \( ax^{2} + bx + c \) is \( x = -\frac{b}{2a} \). Substituting the values, \( x = -\frac{10}{2(-1)} = 5 \). Substituting \( x = 5 \) into the equation \( y = -x^{2} + 10x - 23 \) gives the vertex as \( (5, 2) \).

Step 5: Sketch the Parabola

Plot the vertex \((5, 2)\) and sketch the downward-opening parabola through this vertex and symmetric points.

Step 6: Identify the Solution Region

Since the inequality is \( y > -x^{2} + 10x - 23 \), the solution region is above the parabola. Use a dashed line for the parabola because the inequality is strict (\(>\) not \(\geq\)).

Step 7: Shade the Correct Region

Shade the region above the parabola, as this represents all points \((x, y)\) where \( y > -x^{2} + 10x - 23 \).

Key Concepts

Quadratic Inequality

In the exercise, we are working with a quadratic inequality given by \[ y > -x^{2} + 10x - 23 \]Understanding quadratic inequalities is key in graphing the solution region. A quadratic inequality is an inequality involving a quadratic polynomial, typically expressed in the form \( y > ax^{2} + bx + c \), where \( a \), \( b \), and \( c \) are constants. The expression can change slightly based on the inequality sign, such as \( >, <, \geq, \text{or} \leq \). Quadratic inequalities like these define regions in the coordinate plane rather than a single line or curve. When graphing, one must treat the boundary as an equation first, plot it, and then determine where the inequality holds true. These inequalities often take the form of areas above or below a parabola on a graph. Breaking it down, the given quadratic inequality signifies the area

• Above the parabola defined by the equation \( y = -x^{2} + 10x - 23 \)
• The parabola itself serving as a guide for determining the inequalities that define the region

Parabola

The foundation of graphing a quadratic inequality is the parabola. A parabola is the graph of a quadratic function, which takes the form \( y = ax^{2} + bx + c \). This shape is defined by its curve, and its direction can either be upward or downward depending on the sign of the coefficient \( a \). Here, since \( a = -1 \), the parabola opens downward, creating a hill shape. This can be visualized as an arching structure hovering over the x-axis. A parabola has the following characteristics:

• Symmetry along a vertical axis known as the axis of symmetry.
• A single peak or lowest point called the vertex.
• Extends infinitely in both directions along the axis of symmetry.

Understanding these traits helps in sketching the parabola accurately, which is essential for determining the solution region.

Vertex

Every parabola has a vertex, which is the highest or lowest point depending on the direction the parabola opens. In terms of a quadratic inequality, the vertex serves as a crucial point of reference for sketching.To find the vertex of the parabola represented by \( y = -x^{2} + 10x - 23 \), we use the vertex formula:\[ x = -\frac{b}{2a} \]With \( a = -1 \) and \( b = 10 \), substituting these values into the formula gives:\[ x = -\frac{10}{2(-1)} = 5 \]Plug this back into the equation to find the y-coordinate:\[ y = -(5)^{2} + 10 \times 5 - 23 = 2 \]Thus, the vertex is at the point (5, 2).This point is pivotal as it helps in accurately sketching the parabola and understanding where the inequality holds true.

Solution Region

The solution region for the quadratic inequality is a crucial concept, indicating all the points that satisfy the inequality. Once the boundary parabola is graphed, identifying this region becomes more straightforward. Because our inequality is strict (">"), our parabola will be dashed, indicating that points on the parabola are not included in the solution. In this exercise, since the inequality is given as \( y > -x^{2} + 10x - 23 \), the solution region is:

• Above the parabola, where the values of \( y \) exceed those on the parabola itself.
• Graphically represented by shading the area above the dashed parabola.
• This shading represents all possible solutions that satisfy the inequality.

Recognizing and correctly shading this area is the final step in depicting the solution to the inequality, emphasizing all points (x, y) above the parabola.