Problem 28 - Algebra Form and Function

Question

Identify the slope and \(y\) -intercept and graph the function. $$ f(x)=-2 x+5 $$

Step-by-Step Solution

Answer

Answer: The slope of the linear function is -2 and the y-intercept is 5. To graph the function, plot the y-intercept (0, 5) and use the slope to find the next point (1, 3). Then, draw a line through these two points to create the graph of the function f(x) = -2x + 5.

Step 1: Identify the slope

The slope of the linear function is the coefficient of x in the equation. In the given equation, f(x) = -2x + 5, the coefficient of x is -2. Therefore, the slope is -2.

Step 2: Identify the y-intercept

The y-intercept of the linear function is the constant term in the equation. In the given equation, f(x) = -2x + 5, the constant term is 5. Therefore, the y-intercept is 5.

Step 3: Graph the function

To graph the linear function, we will follow these steps: 1. Plot the y-intercept on the graph: (0, 5) 2. Use the slope to find the next point on the graph. Since the slope is -2, we can go down 2 units and to the right 1 unit from the y-intercept. That will give us the point (1, 3). 3. Draw a line through the two points (0, 5) and (1, 3) to create the complete graph of the linear function f(x) = -2x + 5.

Key Concepts

Slope

The **slope** of a linear function is crucial for determining its steepness and direction. It is represented by the "coefficient of" the variable **x** in the equation of the line. In the equation you're working with, **f(x) = -2x + 5**, the slope is **-2**. This number tells you two key things:

• Direction: A positive slope means the line moves upwards from left to right, while a negative slope, like **-2**, means the line moves downwards.
• Steepness: The absolute value of the slope indicates how steep the line is. A larger number means a steeper line. Here, the line falls **2 units** down for every step **1 unit** horizontally to the right.

This makes it a simple way to quickly understand and visualize the behavior of any linear function. Remember, the slope shows not just how steep the line is but also in which direction it leans.

Y-Intercept

The **y-intercept** is the point at which the line of a linear function crosses the **y-axis**. This is the point where the value of **x** is zero. In the equation **f(x) = -2x + 5**, the y-intercept is **5**.

• Understanding: This term gives you valuable insight into where the line starts on the y-axis before considering the effect of the slope.
• Graphically: On a graph, you'd mark the point **(0, 5)** as the y-intercept. No matter what the slope is, you always begin plotting a line from here.

The y-intercept is easy to spot because it's the constant term in the equation, making it straightforward to find and plot on a graph. Knowing the y-intercept helps in quickly establishing a point on the graph from which other points can be calculated.

Graphing Linear Equations

**Graphing linear equations** involves plotting points on a coordinate plane and drawing a line through them. Here is a simple guide to graph the equation **f(x) = -2x + 5**:

• Plot the Y-Intercept: Begin by plotting the point **(0, 5)** on the y-axis. This is your starting point since x = 0.
• Use the Slope: Knowing the slope is **-2**, move **2 units** down and **1 unit** to the right from the y-intercept. This movement gives you a new point **(1, 3)**. You can keep applying this method to find more points if needed.
• Draw the Line: Once you have at least two points—(0, 5) and (1, 3)—use a ruler to draw a straight line through them. Extend the line in both directions and add arrows to show that it continues indefinitely.

This technique makes graphing linear functions manageable and straightforward. With practice, graphing can help visualize how changes in the equation affect the line's position and angle on the graph.