Problem 12 - Algebra 2

Question

Graph each inequality. $$ y<-x^{2}+7 x+8 $$

Step-by-Step Solution

Answer

Graph the parabola boundary with a dashed line and shade below it.

Step 1: Identify the Inequality Type

The given inequality is quadratic, due to the presence of the term \(x^2\). Quadratic inequalities typically have parabolic graphs.

Step 2: Rewrite the Inequality as Equation

To find the boundary of the region, rewrite the inequality \(y < -x^2 + 7x + 8\) as the equation \(y = -x^2 + 7x + 8\). This equation represents the parabola.

Step 3: Determine the Vertex

For a quadratic equation of the standard form \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be found using \(x = -\frac{b}{2a}\). Here, \(a = -1\) and \(b = 7\), so the vertex's x-coordinate is \(x = -\frac{7}{2 \times -1} = \frac{7}{2}\). Substitute this back into the equation to find the y-value of the vertex.

Step 4: Calculate the Y-Coordinate of the Vertex

Substitute \(x = \frac{7}{2}\) into the equation \(y = -x^2 + 7x + 8\) to find the y-coordinate of the vertex: \( y = -\left(\frac{7}{2}\right)^2 + 7\left(\frac{7}{2}\right) + 8 \) After solving, you will get the y-coordinate.

Step 5: Find Additional Points

Choose a few x-values around the vertex to find additional points on the parabola, such as when \(x = 0\) and \(x = 1\). Substitute these x-values into the equation \(y = -x^2 + 7x + 8\) to find corresponding y-values.

Step 6: Graph the Parabola

Plot the vertex along with the additional points on a coordinate plane. Draw a dashed curve through these points to indicate the boundary is not included (as the inequality is '<').

Step 7: Shade the Appropriate Region

Since the inequality is \(y < -x^2 + 7x + 8\), shade the region below the parabola to indicate all the points where \(y\) is less than the boundary.

Key Concepts

Parabolic Graphs

Quadratic inequalities, like the one given in the problem, are often represented graphically using a parabolic shape. A parabola is a U-shaped curve that is symmetrical and can open either up or down. In this specific case, we have the inequality \( y < -x^2 + 7x + 8 \). The negative coefficient of \( x^2 \) indicates that the parabola opens downward. This means the curve will have a maximum point known as the vertex.
Understanding the direction in which the parabola opens helps us predict the overall behavior of the graph, especially when graphing inequalities.

Vertex Calculation

The vertex of a parabola plays a crucial role in understanding the graph's shape and position. For a quadratic function given in standard form \( y = ax^2 + bx + c \), the x-coordinate of the vertex is calculated using the formula \( x = -\frac{b}{2a} \). In our inequality, \( a = -1 \) and \( b = 7 \), giving us an x-coordinate of \( x = \frac{7}{2} \).
To find the full vertex, we substitute \( x = \frac{7}{2} \) back into the equation as follows:

• Insert \( x = \frac{7}{2} \) into the equation: \( y = -\left(\frac{7}{2}\right)^2 + 7\left(\frac{7}{2}\right) + 8 \)
• Calculate the resulting y-coordinate. This completes the vertex \((\frac{7}{2}, y)\).

The vertex is a key point to plot on the graph and helps in visualizing the parabola's path.

Graphing Inequalities

Graphing inequalities involves more than just plotting points; it requires shading regions to convey solutions. Begin by sketching the boundary, in this case, the parabola described by \( y = -x^2 + 7x + 8 \). However, since our inequality is \( y < -x^2 + 7x + 8 \), we use a dashed line to graph the parabola, indicating that points on the boundary are not included in the solution set.
Once the parabola is drawn, we need to shade the area beneath it. This shaded region represents all the solutions where the inequality holds true, meaning for any point in this region, the y-values will indeed be less than the expression \(-x^2 + 7x + 8\).

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically represented by \( y = ax^2 + bx + c \). These functions create curved graphs known as parabolas. Quadratics are pivotal in breaking down and analyzing parabolic graphs within inequalities.
Recognizing the standard form \( ax^2 + bx + c \) helps with identifying key attributes such as the vertex, axis of symmetry, and direction of opening. In the exercise, the presence of \( x^2 \) confirms the quadratic nature, while the coefficients give insight into graph characteristics, such as:

• The sign of \( a \) (in this case, negative) dictates whether the graph opens upwards or downwards.
• The vertex is calculated using specific formulas derived from the coefficients \( b \) and \( a \).

Understanding these components aids in graphing and solving the inequalities effectively.