Question

Sketch the graph of the inequality. $$y \geq-4 x^{2}-3 x+8$$

Short answer

The short answer would be, the parabola of equation \(y=-4x^2-3x+8\), which opens downwards. The solution to the inequality \(y \geq-4 x^{2}-3 x+8\) including all the points above and on the parabola.

Step-by-step solution

Understanding the equation

The equation \(y=-4x^2-3x+8\) is a quadratic function. Its graph will be a parabola. The coefficient of \(x^2\) is negative, so the parabola will open downwards.

Find the vertex of the parabola

The vertex of the parabola given by the equation \(y=ax^2+bx+c\) can be found using the formula \((-\frac{b}{2a} , f(-\frac{b}{2a}))\). So, for the given equation where \(a=-4\) and \(b=-3\), the x-coordinate of the vertex is \(-\frac{b}{2a} = -\frac{-3}{2*-4} = 0.375\). Now put this x-coordinate value into our equation to get the y-coordinate: \(y=-4*(0.375)^2 -3*0.375 +8 =7.5625\). So, the vertex is at (0.375,7.5625).

Identify other points on the parabola

In addition to the vertex, we need at least one more point to graph the parabola. Let's find the intercepts. The y-intercept is the point where \(x=0\). So, let's calculate \(y = -4*0^2 -3*0 + 8 = 8\). Therefore, the y-intercept is (0,8). The x-intercept(s) are calculated by setting \(y=0\) and solving for \(x\). So, \(0 = -4x^2 - 3x + 8 \) is the equation to solve. Factoring, we get \(0 = -(4x - 5)(x + 2)\). Therefore, our x-intercepts are \(\frac{5}{4} = 1.25\) and -2. Thus, the x-intercepts are (1.25,0) and (-2,0).

Sketch the graph

In this part plotting the vertex and the determined points to form the curve of the parabola on a graph. A parabola symmetric so make certain that it appears this way on the graph.

Shade the correct region

The inequality symbol is \(\geq\), this means we will shade the region that includes all y-values that are greater than or equal to our parabola. Since the parabola opens downwards, we shade the region above the parabola curve to represent all the solutions of our inequality.

Key concepts

Quadratic Function

Understanding a quadratic function is essential when dealing with parabolas and their respective inequalities. In its standard form, a quadratic function is represented as
\( y = ax^2 + bx + c \),
where \( a \), \( b \), and \( c \) are constants. Here's what each component means:

• \( a \) determines the direction of the parabola's opening (upwards if positive, downwards if negative).
• \( b \) influences the parabola's position laterally on the x-axis.
• \( c \) is the y-intercept, the point where the parabola crosses the y-axis.

When you come across the inequality like \( y \geq -4x^2 - 3x + 8 \), it is signaling the region on the graph where the y-values are greater than or equal to the values of the quadratic function. The graph of such an inequality indicates a set of all possible solutions that satisfy the condition.

Vertex of a Parabola

The vertex of a parabola is a key feature, representing its highest or lowest point, depending on whether it opens up or down. For the quadratic function \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula
\( \text{x-coordinate} = -\frac{b}{2a} \),
and you can obtain the y-coordinate by plugging this x back into the original equation. For instance, with the equation \( -4x^2 - 3x + 8 \), the vertex would be calculated as
\( (0.375, 7.5625) \).
Knowing the vertex is essential for understanding the parabola's shape and direction, and for sketching its graph accurately. The vertex also helps determine the axis of symmetry, which is a straight line that divides the parabola into two mirror-image halves.

X-Intercepts and Y-Intercepts

Intercepts are where the graph of the function crosses the axes. A y-intercept is where the parabola meets the y-axis (when \( x=0 \)). For the quadratic function given, the y-intercept is (0,8), which you get by setting \( x \) to zero in the equation.

The x-intercepts, on the other hand, are points where the graph crosses the x-axis (when \( y=0 \)). They are found by solving the equation \( -4x^2 - 3x + 8 = 0 \). By factoring or using the quadratic formula, you get the x-intercepts (1.25,0) and (-2,0). These points are crucial for graphing since they provide a clear starting point to begin sketching the curve of a parabola, especially when visualizing the impact of inequalities on a coordinate plane.