Function Analysis: Coefficients, Zeros, And Graphs

Hey everyone! Let's dive into some math fun, specifically focusing on functions! We're going to break down two functions, figuring out their secrets like the coefficients, zeros, and how they behave on a graph. Buckle up, because it's going to be a ride!

Understanding the Basics: Linear Functions

Before we jump into the functions, let's refresh our memories on the basics. We're dealing with linear functions here. These functions are super cool because their graphs are straight lines. A linear function is generally written as f(x) = mx + b, where:

• m is the angular coefficient (or slope). It tells us how steep the line is and in which direction it's going (up or down).
• b is the linear coefficient (or y-intercept). It's the point where the line crosses the y-axis (where x = 0).

So, when you see a linear function, you instantly know it's a straight line, and you can easily figure out its slope and where it crosses the y-axis just by looking at the equation. Pretty neat, huh?

Now, let's get our hands dirty with the functions we've been given.

Function 1: f(x) = -x + 3

(a) Coefficients: Unveiling the Slope and Intercept

Alright, let's analyze the first function, f(x) = -x + 3. To find the coefficients, we just need to compare this function to our general form f(x) = mx + b. See, it's like a secret code!

• The angular coefficient (m) is the number multiplying x. In this case, it's -1 (since we can rewrite the function as f(x) = -1x + 3). This means our line has a negative slope. When the x value increases, the y value decreases.
• The linear coefficient (b) is the constant term, which is +3. This tells us the line crosses the y-axis at the point (0, 3).

So, we've found that this line has a negative slope (going downwards from left to right) and crosses the y-axis at y = 3. Cool, right?

(b) Increasing or Decreasing: The Slope's Tale

Now, is this function increasing or decreasing? This is super easy! We look at the angular coefficient (m). Since m = -1, which is negative, the function is decreasing. This means as x increases, f(x) gets smaller and smaller. It's like going down a hill.

(c) Zero of the Function: Where the Line Hits Zero

The zero of the function is the x-value where f(x) = 0. In other words, it's where the line crosses the x-axis. To find it, we need to solve the equation -x + 3 = 0. Let's do it:

• -x + 3 = 0
• -x = -3
• x = 3

So, the zero of the function is x = 3. This means the line crosses the x-axis at the point (3, 0).

(d) Graph of the Function: Visualizing the Line

To graph this function, we can use the information we've already gathered:

• Y-intercept: The line crosses the y-axis at (0, 3).
• Zero (x-intercept): The line crosses the x-axis at (3, 0).
• Slope: The line has a negative slope.

Plot these two points on a graph (0, 3) and (3, 0), and then draw a straight line through them. That's your graph! You'll see a line going down from left to right, crossing the y-axis at 3 and the x-axis at 3.

(e) Study of the Function: Summing It Up

Let's recap what we've learned about f(x) = -x + 3:

• It's a linear function.
• Its angular coefficient is -1, and its linear coefficient is 3.
• It's a decreasing function.
• Its zero is x = 3.
• Its graph is a straight line going down from left to right, crossing the y-axis at 3 and the x-axis at 3.

Function 2: f(x) = 4x + 2

(a) Coefficients: Deciphering the Equation

Let's move on to our second function, f(x) = 4x + 2. Again, we compare it to f(x) = mx + b:

• The angular coefficient (m) is 4. This means our line has a positive slope. For every unit increase in x, f(x) increases by 4.
• The linear coefficient (b) is 2. This means the line crosses the y-axis at the point (0, 2).

(b) Increasing or Decreasing: The Slope's Verdict

Since the angular coefficient (m) is 4 (positive), the function is increasing. This means as x increases, f(x) also increases. It's like going up a hill.

(c) Zero of the Function: Finding the X-Intercept

To find the zero, we solve 4x + 2 = 0:

• 4x + 2 = 0
• 4x = -2
• x = -2/4
• x = -1/2

So, the zero of the function is x = -1/2. The line crosses the x-axis at the point (-1/2, 0).

(d) Graph of the Function: Bringing It to Life

To graph this function:

• Y-intercept: The line crosses the y-axis at (0, 2).
• Zero (x-intercept): The line crosses the x-axis at (-1/2, 0).
• Slope: The line has a positive slope.

Plot these points (0, 2) and (-1/2, 0), and draw a straight line through them. This line will go up from left to right, crossing the y-axis at 2 and the x-axis at -1/2.

(e) Study of the Function: Key Takeaways

Let's summarize f(x) = 4x + 2:

• It's a linear function.
• Its angular coefficient is 4, and its linear coefficient is 2.
• It's an increasing function.
• Its zero is x = -1/2.
• Its graph is a straight line going up from left to right, crossing the y-axis at 2 and the x-axis at -1/2.

Wrapping Up: Functions Explained!

Awesome, guys! We've successfully analyzed two linear functions, figuring out their coefficients, zeros, and how they look on a graph. Remember, the angular coefficient tells you about the slope and direction, the linear coefficient gives you the y-intercept, and the zero helps you find where the line crosses the x-axis. Keep practicing, and you'll become function masters in no time! Remember to always try to use the general form f(x) = mx + b as your secret weapon! See you next time, and keep exploring the amazing world of math!