Problem 9 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the inequality. $$ y \leq x^{2} $$

Step-by-Step Solution

Answer

The graph plotted is an upward facing parabola of the function \(y = x^{2}\). The entire region below this curve, including the parabola is shaded, representing the solution to the inequality \(y \leq x^{2}\).

Step 1: Understand the function

First, we need to understand the function to be graphed. The function \(y = x^{2}\) is a quadratic function which opens upwards. Sketch this by thinking of some x values and corresponding y values, e.g. (-2, 4), (-1, 1), (0, 0), (1, 1), and (2, 4). This forms a U-shape.

Step 2: Graph the function

Plot those points and sketch a curve that passes through them. This would be a parabola that opens upwards with a minimum point (0, 0).

Step 3: Graph the inequality

The inequality \(y \leq x^{2}\) involves all the points on the graph of \(y = x^{2}\) as well as all the points below it since y values are less than or equal to the x-values squared. So, shade the entire region below the curve including the curve itself, as these points satisfy the inequality.

Key Concepts

Understanding Quadratic Functions

A quadratic function is represented in the form of \( y = ax^2 + bx + c \). In this case, we are dealing with \( y = x^2 \), which is a special type of quadratic function with \( a = 1 \), \( b = 0 \), and \( c = 0 \). The graph of any quadratic function is a parabola. The quadratic function \( y = x^2 \) is one of the simplest forms and serves as the basis for understanding all quadratic graphs.
This function has:

• A single variable raised to the power of 2 (i.e., \( x^2 \)).
• No linear term or constant term (i.e., \( b = 0 \) and \( c = 0 \)).

The vertex of this particular function is at the origin, (0, 0). The coefficient "a" determines the direction of the parabola. Since \(a = 1\), which is positive, the parabola opens upwards. Understanding these properties helps us sketch the graph precisely.

Characteristics of Parabolas

Parabolas are U-shaped curves that can open either upwards or downwards depending on the sign of the coefficient in front of \(x^2\) in the function. For \( y = x^2 \), the parabola opens upwards owing to the positive coefficient.Parabolas have several key features:

• Vertex: The turning point of the parabola. For \( y = x^2 \), this is at (0, 0).
• Axis of Symmetry: A vertical line that splits the parabola into two mirror-image halves. In this case, it is the line \( x = 0 \).
• Direction: As mentioned, if \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
• Intersection with the y-axis: The point where the graph crosses the y-axis, which occurs at the origin here since \( c = 0 \).

Sketching these features helps create an accurate representation of the graph. Given these properties, we can understand that a parabola illustrates the shape of quadratic functions effectively.

Shading Regions of Inequalities

When graphing inequalities such as \( y \leq x^2 \), beyond drawing the parabola itself, we need to consider the region that satisfies the inequality. Here, it is not just the curve we care about; it's all the points that lie either on or below the parabola since the inequality is "less than or equal to."
Here's a simple way to determine shading:

• First, draw the parabola for \( y = x^2 \).
• Then, since the inequality is \( \leq \), the shading includes all points on the curve and those beneath it.

The reason we shade below the parabola is that for any point \((x, y)\), the equation \( y \leq x^2 \) holds true for all "y" values less than or equal to \( x^2 \). When graphing, shading correctly is crucial for visually representing all possible solutions to the inequality.