Graph each inequality. y > 2^x + 1

Verified step by step guidance

1. Identify the boundary curve by rewriting the inequality as an equation: \(y = 2^{x} + 1\). This curve will help define the region for the inequality.
2. Graph the curve \(y = 2^{x} + 1\). Since \$2^{x}\( is an exponential function, the graph will rise rapidly as \)x\( increases and approach \)y = 1\( as \)x$ goes to negative infinity.
3. Determine the type of boundary line. Because the inequality is strict (\(y > 2^{x} + 1\)), draw the curve as a dashed line to indicate points on the line are not included in the solution.
4. Choose a test point not on the boundary (commonly \((0,0)\)) to check which side of the curve satisfies the inequality. Substitute \(x=0\) and \(y=0\) into \(y > 2^{x} + 1\) to see if the inequality holds.
5. Shade the region where the inequality is true based on the test point result. If the test point satisfies the inequality, shade the side containing that point; otherwise, shade the opposite side.

Key Concepts

Exponential Functions

An exponential function has the form y = a^x, where the variable is in the exponent. In this question, y = 2^x + 1 shifts the basic exponential graph y = 2^x upward by 1 unit. Understanding the shape and behavior of exponential functions is essential for graphing the inequality.

Graphing Inequalities

Graphing inequalities involves shading the region of the coordinate plane that satisfies the inequality. For y > 2^x + 1, you first graph the boundary curve y = 2^x + 1, then shade the area above it because y is greater than the function values.

Boundary Lines and Open/Closed Curves

When graphing inequalities, the boundary line or curve is drawn solid if the inequality includes equality (≥ or ≤) and dashed if it does not (> or <). Since the inequality is y > 2^x + 1, the boundary curve y = 2^x + 1 is dashed, indicating points on the curve are not included.