Graph each inequality. y>2^x

Verified step by step guidance

1. Identify the inequality to graph: \(y > 2^{x}\). This means we want to graph all points where the \(y\)-value is greater than \$2^{x}$.
2. First, graph the boundary curve \(y = 2^{x}\). This is an exponential function where the base is 2, so it passes through points like \((0,1)\) since \$2^{0} = 1\(, and increases as \)x$ increases.
3. Since the inequality is strict (\(>\)), draw the curve \(y = 2^{x}\) as a dashed line to indicate that points on the line are not included in the solution.
4. Determine which side of the curve to shade by testing a point not on the curve, such as \((0,0)\). Substitute into the inequality: \$0 > 2^{0}\( becomes \)0 > 1$, which is false, so do not shade below the curve.
5. Shade the region above the curve \(y = 2^{x}\), representing all points where \(y\) is greater than \$2^{x}$.

Key Concepts

Exponential Functions

An exponential function has the form y = a^x, where the base a is a positive constant. In this question, y = 2^x represents an exponential growth function, which increases rapidly as x increases. Understanding its shape and behavior is essential for graphing related inequalities.

Graphing Inequalities

Graphing an inequality like y > 2^x involves shading the region of the coordinate plane where the inequality holds true. The boundary curve y = 2^x is graphed first, and since the inequality is strict (greater than), the boundary is drawn as a dashed line to indicate points on the curve are not included.

Coordinate Plane and Regions

The coordinate plane is divided by the graph of the function into regions. For y > 2^x, the region above the curve is shaded. Understanding how to identify and shade the correct region based on the inequality symbol is crucial for accurately representing the solution set.