Problem 36 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The graphical solution to the inequality \(y \leq -x^2 + 3x + 4\) is a downward-opening parabola with vertex at (1.5, 5.25) and y-intercept at \(y = 4\). The solution to the inequality includes all points below or on this parabola, represented by shading in this area on the graph.

Step 1: Function recognition

Based on the equation \(y = -x^2 + 3x + 4\), it is a quadratic function. The standard form of a quadratic function is \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Here, \(a = -1\), \(b = 3\), and \(c = 4\). The graph of this function will be a parabola that opens downward because the coefficient of \(x^2\) is negative.

Step 2: Sketching the function

To sketch the graph of the function, first find the vertex, which is at \((h, k)\) where \(h = -b/2a\) and \(k = f(h)\). Here, the vertex is at \((1.5, 5.25)\). Then, because the parabola opens downward, draw the graph accordingly. The y-intercept is at \(y = 4\), so mark this point. The last step is to draw the parabola with these key points.

Step 3: Determining the solution region

The inequality is \(y \leq -x^2 + 3x + 4\). This means that the solution to the problem includes all points below or on the graph of the function described by the equation \(y = -x^2 + 3x + 4\). To represent this, shade in the region of the graph below the parabola.

Key Concepts

Quadratic Function

When we come across a mathematical expression in the form of y = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0, we're dealing with a quadratic function. These functions are fundamental in the study of algebra and play a crucial role in various applications, from physics to finance.

A key characteristic of this function is the shape of its graph: a smooth curve called a parabola. Depending on the sign of the leading coefficient—the a value—the parabola will either open upwards (a > 0) or downwards (a < 0). In our exercise example, since the equation is y = -x^2 + 3x + 4, we can identify that the parabola opens downward because the coefficient of x^2 is negative (a = -1).

Understanding the direction in which the parabola opens is crucial as it affects the graph's range and the location of the maximum or minimum point—the vertex of the parabola.

Vertex of a Parabola

The vertex of a parabola is a significant point where the curve changes direction, which makes it either the highest or lowest point on the graph. For a quadratic function y = ax^2 + bx + c, the vertex (h, k) can be found using the formula h = -b/2a and then evaluating the function at h to find k.

In our example, the quadratic function is y = -x^2 + 3x + 4. Calculating the vertex using h = -b/2a, we substitute b = 3, and a = -1 to get h = 1.5. Finding k then involves substituting h back into the function yielding k = 5.25, placing the vertex at (1.5, 5.25). The vertex not only helps to graph the quadratic function accurately but also plays an important role in understanding the function's optimal values and the effect of any transformations applied to the parabola.

Solution Region of Inequality

Graphing an inequality like y ≤ -x^2 + 3x + 4 is about visualizing the set of all possible solutions. In the context of quadratic inequalities, the solution region can be either above or below the parabola, depending on the direction of the inequality symbol. If we have y ≤ (quadratic expression), the solution region is the space on the coordinate plane that lies on or below the parabola. Conversely, for y ≥ (quadratic expression), it's the space on or above the parabola.

After sketching the parabola of the quadratic function, which serves as the boundary of our inequality, the solution region is shaded to represent all the (x, y) pairs that satisfy the inequality. In the exercise example, we shaded the region below the parabola because our inequality is y ≤ -x^2 + 3x + 4. This shaded area includes truly innumerable points, each representing a valid solution to the inequality, giving a visual and intuitive understanding of the solutions to the inequality.