The linear function f(x)=2x+3 is in the form f(x)=mx+b Use the sliders to change the parameters of m and b Observe how the key features of the function change as each parameter is adjusted. Next, open the Notebook tool (B), and write down your observations based on the changes made to the parameters and your knowledge of parent functions. +MARK + RELATIONSHIP - + DATA () y=mx+b x v m: -2.8 -10 10 b: 3 -10 10

When analyzing the linear function \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, adjusting these parameters affects the graph of the function in specific ways:<br /><br />1. **Slope (\( m \))**:<br /> - The slope \( m \) determines the steepness and direction of the line.<br /> - If \( m > 0 \), the line slopes upwards from left to right, indicating a positive relationship between \( x \) and \( y \).<br /> - If \( m < 0 \), the line slopes downwards from left to right, indicating a negative relationship.<br /> - As the absolute value of \( m \) increases, the line becomes steeper. Conversely, as it decreases, the line becomes flatter.<br /><br />2. **Y-Intercept (\( b \))**:<br /> - The y-intercept \( b \) is the point where the line crosses the y-axis.<br /> - Changing \( b \) shifts the line up or down without altering its slope.<br /> - Increasing \( b \) moves the line upward, while decreasing \( b \) moves it downward.<br /><br />### Observations Based on Changes:<br /><br />- **Changing \( m \)**:<br /> - With \( m = -2.8 \), the line will have a steep negative slope, indicating a strong inverse relationship between \( x \) and \( y \). The line will descend sharply from left to right.<br /><br />- **Changing \( b \)**:<br /> - With \( b = 3 \), the line will intersect the y-axis at \( (0, 3) \). Adjusting \( b \) would move this intersection point vertically along the y-axis.<br /><br />### Key Features of Linear Functions:<br /><br />- **Linearity**: The graph is always a straight line.<br />- **Constant Rate of Change**: The slope \( m \) represents a constant rate of change; for every unit increase in \( x \), \( y \) changes by \( m \) units.<br />- **No Curvature**: Unlike quadratic or higher-degree polynomial functions, linear functions do not curve.<br /><br />By using sliders to adjust \( m \) and \( b \), you can visually observe these effects on the graph, reinforcing the understanding of how each parameter influences the linear function's behavior.