Problem 3 - Algebra 2

Question

Graph each inequality. $$ y > -2 x^{2}-4 x+3 $$

Step-by-Step Solution

Answer

Shade the region above the parabola defined by \( y = -2x^2 - 4x + 3 \).

Step 1: Identify the inequality and corresponding function

The inequality given is \( y > -2x^2 - 4x + 3 \). This implies that we need to investigate the quadratic function \( y = -2x^2 - 4x + 3 \). Since the inequality symbol is '>', we are interested in the region above this parabola.

Step 2: Determine the vertex of the parabola

The vertex form of a parabola is useful for graphing. The vertex \((h, k)\) of a parabola in standard form \( ax^2 + bx + c \) can be calculated using the formula \( h = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = -4 \). Hence, \( h = -\frac{-4}{2(-2)} = 1 \). To find \( k \), substitute \( x = 1 \) into the equation \( y = -2x^2 - 4x + 3 \). We find \( k = -2(1)^2 - 4(1) + 3 = -3 \). So, the vertex is \( (1, -3) \).

Step 3: Determine the axis of symmetry

The axis of symmetry of the parabola is the vertical line passing through the vertex. For this parabola, the axis of symmetry is \( x = 1 \).

Step 4: Plot the parabola

Using the vertex \((1, -3)\) and the fact that the parabola opens downwards (since \( a = -2 < 0 \)), plot at least a few points on either side of the vertex to sketch the parabola. Choose values of \( x \) such as 0, 2, -1, and find the corresponding \( y \) values to plot the curve.

Step 5: Shade the region above the parabola

Since the inequality is \( y > -2x^2 - 4x + 3 \), we shade the region above the parabola to represent all the points where \( y \) is greater than the value of the quadratic function. Do not include the parabola itself, as this is a strict inequality ('>').

Key Concepts

Graphing Inequalities

Graphing inequalities involves not just plotting a line or curve but also determining the area that satisfies the inequality condition. When you have an inequality such as \( y > -2x^2 - 4x + 3 \), you're essentially looking for all the points where \( y \) is greater than the quadratic expression. To graph this:

• First, plot the corresponding equation \( y = -2x^2 - 4x + 3 \) to find the parabola.
• Use the inequality sign to determine which region to shade. Here, since it's a 'greater than' inequality, you'd shade above the curve.
• Remember not to include the line of the parabola itself since the inequality is strict \( (>) \).

By shading the appropriate region, you are showing all possible solutions for the inequality. This visual representation helps in understanding where the values of \( y \) make the inequality true.

Quadratic Functions

Quadratic functions are essential concepts in algebra represented as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The graph of a quadratic function is a parabola. Unlike linear functions creating straight lines, quadratic functions curve due to the \( x^2 \) term.The parabola can open upwards or downwards:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

In the exercise, \( a = -2 \), so the parabola opens downwards. Understanding this helps predict the shape and direction of the parabola before graphing. A downward opening parabola implies that as \( x \) moves away from the vertex in either direction, \( y \) decreases.

Vertex Form of a Parabola

The vertex form of a parabola provides a clear way to visualize and graph quadratic functions. While the standard form is \( ax^2 + bx + c \), the vertex form is \( a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.To interpret from standard to vertex form, determine the vertex using the formula:

• \( h = -\frac{b}{2a} \)
• \( k \) is found by substituting \( h \) back into the function to get the \( y \)-coordinate of the vertex.

In our exercise, \( a = -2 \) and \( b = -4 \), leading to \( h = 1 \) and, by substituting this into the equation, \( k = -3 \). So, the vertex is \((1, -3)\). The vertex is crucial as it provides the highest or lowest point on the graph, guiding the symmetry and the shape of the parabola. This symmetry makes graphing easier once the vertex and direction are known.