Problem 30 - Algebra 2

Question

Graph each inequality. $$ y \geq-x^{2}-7 x+10 $$

Step-by-Step Solution

Answer

Graph the quadratic function and shade the region above the parabola with a solid line to include the boundary.

Step 1: Identify the inequality type

The given inequality is \( y \geq -x^2 - 7x + 10 \). This represents a quadratic inequality where the quadratic expression is in terms of \( x \). The inequality sign \( \geq \) means that the region above or on the parabola should be shaded.

Step 2: Determine the vertex and axis of symmetry

The inequality is in the form \( y \geq ax^2 + bx + c \), where \( a = -1 \), \( b = -7 \), and \( c = 10 \). The vertex \((h, k)\) of the parabola can be found using the vertex formula \( h = -\frac{b}{2a} \). Substituting the values gives \( h = -\frac{-7}{2(-1)} = \frac{7}{2} \approx 3.5 \). Substitute \( x = 3.5 \) back into the equation to find \( k \): \( k = -(3.5)^2 - 7(3.5) + 10 = -12.25 - 24.5 + 10 \approx -26.75 \). So, the vertex is approximately \( (3.5, -26.75) \).

Step 3: Find the roots (if needed)

To find where the graph intersects the x-axis (roots), set \( -x^2 - 7x + 10 = 0 \) and solve for \( x \). Apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plug the values into the formula: \( x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(-1)(10)}}{2(-1)} \). Simplifying gives \( x = \frac{7 \pm \sqrt{49 + 40}}{-2} = \frac{7 \pm \sqrt{89}}{-2} \), which are the roots of the quadratic.

Step 4: Sketch the parabola

The parabola opens downwards because the coefficient of \( x^2 \), \( a = -1 \), is negative. Plot the vertex \((3.5, -26.75)\) and the roots \( x = \frac{7 + \sqrt{89}}{-2} \) and \( x = \frac{7 - \sqrt{89}}{-2} \) onto a graph. Draw a dotted or solid line through these points. Since the inequality sign is \( \geq \), use a solid line to indicate that points on the graph are included.

Step 5: Shade the correct region

Since the inequality is \( y \geq -x^2 - 7x + 10 \), shade above the parabola to indicate the solution set where \( y \) values are greater than or equal to the quadratic expression. This shading indicates all points \((x, y)\) that satisfy the inequality.

Key Concepts

Vertex of a Parabola

In a quadratic graph, the vertex is a significant point. It's where the parabola changes direction. If the parabola opens upwards, the vertex is the lowest point. If it opens downwards, as in this case, it's the highest point. The vertex of a parabola described by the equation \[ y = ax^2 + bx + c \]can be found using the formula for the vertex, which includes \[ h = -\frac{b}{2a} \] for the x-coordinate.

• In our exercise, \( a = -1 \) and \( b = -7 \).
• Plug these into the equation to get \( h = -\frac{-7}{2(-1)} = \frac{7}{2} \), approximately 3.5.

To obtain the y-coordinate (k), substitute the x-coordinate into the original equation:
\[ k = -h^2 - 7h + 10 \]Substitute \( h = 3.5 \) to find \( k \):
\[ k = -(3.5)^2 - 7(3.5) + 10 \approx -26.75 \]The vertex is then approximately \((3.5, -26.75)\). It acts as the marker for graphing and helps you understand how the parabola sits on the coordinate plane.

Axis of Symmetry in Quadratic Equations

The axis of symmetry is an imaginary line that vertically slices the parabola into two equal halves. For a quadratic equation of the form \[ y = ax^2 + bx + c \]the axis of symmetry can be described by the equation \[ x = -\frac{b}{2a} \].

• This x-value is identical to the x-coordinate of the vertex.

In our graphing problem, the axis of symmetry is \( x = 3.5 \). This line is pivotal because it shows the balance of the parabola — each side is a mirror reflection of the other.
Understanding this helps give you a visual guide for plotting points equal in distance from the vertex on either side.

Shading Solution Region

Shading the location on a graph is essential when solving inequalities. This step highlights all the potential solutions for the inequality. For \[ y \geq ax^2 + bx + c \],shade the region above the parabola, indicating where the \( y \)values are greater than the calculated quadratic expression.

• If the inequality includes \( \geq \) or \( \leq \), use a solid line.
• For less than or greater than, use a dashed line.

In the given exercise, you will shade the area over the drawn parabola because it's \( \geq \), meaning any point on or above the curve meets the original inequality. This visualization makes it easier to see all possible solutions.

Roots of a Quadratic Equation

Roots of the quadratic equation are the x-values where the parabola intersects the x-axis. These points are crucial for complete graphing. Set the quadratic equation equal to zero to find these roots: \[ ax^2 + bx + c = 0 \].Apply the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

• For this problem: \( a = -1, b = -7, c = 10 \).
• Insert these into the formula to find \( x = \frac{7 \pm \sqrt{89}}{-2} \).

The solution \( \approx \) tells you where to plot the x-intercepts on your graph.
These roots give vital clues about where the graph crosses the x-axis and are instrumental in sketching the curve.