Problem 25 - Algebra 1

Question

Sketch the graph of the inequality. $$y \geq x^{2}-5 x$$

Step-by-Step Solution

Answer

The vertex of the parabola \(y = x^{2}-5x\) is at (2.5, -2.5). The solution set to the inequality \(y \geq x^{2}-5x\) includes all the points on the parabola and the area above it.

Step 1 Identify and Graph the Quadratic Function

Recognize that the equation \(y = x^{2}-5x\) describes a parabola. To sketch this graph, it's helpful to find its vertex first. \nThe vertex form of a quadratic function is \(y = a(x-h)^{2} + k\), where \((h, k)\) are the coordinates of the vertex. \nFirst let's find the x-coordinate of the vertex. It's given by the formula \(h = -\frac{b}{2a}\), where \(a\) and \(b\) are coefficients in the quadratic equation \(ax^{2}+bx+c\). For our case \(a = 1\) and \(b = -5\), therefore \(h = -\frac{-5}{2*1} = \frac{5}{2} = 2.5\). \nThen replace \(h\) in our equation to find \(k\), the y-coordinate of the vertex. \(k = (2.5)^{2}-5*2.5 = -2.5\). \nSo, the vertex of the parabola is at (2.5, -2.5). Sketch the parabolic graph based on this vertex.

Step 2 Identify the Solution for the Inequality

The inequality \(y \geq x^{2}-5x\) implies that the solution is the set of all points above the parabola, including the parabola itself. This means all the area that is above the parabola should be shaded.

Step 3 Sketch the Solution for the Inequality

Sketch the solution of inequality on the coordinate plane where we sketched the parabola in step 1. Again, recall that the vertex is at (2.5, -2.5), and remember to include that point. Then shade the area above the parabola to represent all points that satisfy the inequality \(y \geq x^{2}-5x\). Always include the original parabolic line as part of the solution.

Key Concepts

Quadratic Inequalities

Understanding quadratic inequalities is essential for sketching their graphs and solving quadratic problems. A quadratic inequality, such as \(y eq x^2 - 5x\), represents a region on the coordinate plane where the inequality holds true. Unlike a quadratic equation, which equals a value, an inequality shows a relationship that is either greater than, less than, or including the boundary line.

To solve these, we first handle them as if they were equations to find their curve or boundary line. In our example, \(y = x^2 - 5x\) forms a parabola. Once we graph this, we determine where the inequality is satisfied. For \(y eq x^2 - 5x\), we look for the region either above or below the parabola, depending on the direction of the inequality symbol.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function provides a direct way to identify key features of a parabola. Written as \(y = a(x-h)^2 + k\), it reveals the coordinates of the vertex, \((h, k)\), and gives immediate insight into the shape and direction of the parabola based on the value of \(a\). If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards. Converting a standard quadratic \(ax^2 + bx + c\) into vertex form can be done through the process of completing the square. The benefits of this form are especially clear when graphing, as it allows for quick placement of the vertex and understanding of how the parabola stretches or shrinks relative to the unit parabola \(y = x^2\).

Coordinates of the Vertex

The vertex of a parabola is a point that represents its maximum or minimum value, and it's crucial for sketching the graph accurately. To find the coordinates of the vertex, \((h, k)\), in the vertex form, \(h\) is derived by using the formula \(h = -\frac{b}{2a}\) when dealing with the standard form, \(ax^2 + bx + c\).

In our case, where the standard form is \(y = x^2 - 5x\), with \(a = 1\) and \(b = -5\), we compute \(h = 2.5\). Then, we substitute \(x = 2.5\) back into the equation to find \(k = -2.5\), giving us the vertex coordinates of \((2.5, -2.5)\). Knowing the vertex is vital, as it is a turning point and helps in correctly sketching the parabola and determining the region of inequality.