Question

The linear function \(y=g(x)\) is graphed in the \(x y\)-plane. If \(g(-3)=4\) and \(g(2)=19\), what is the slope of line \(g\) ?

Short answer

The slope of line g is 3.

Step-by-step solution

Identify the coordinates of the two points

We are given two points on the line, which we can label as point A and point B with their corresponding coordinates: Point A: (-3, 4) Point B: (2, 19)

Apply the slope formula

Use the coordinates of the two points identified in Step 1 to find the slope of the line g(x) using the formula: Slope (m) = (y2 - y1) / (x2 - x1)

Plug in the coordinates and calculate

Now, let's plug in the coordinates of points A and B into the slope formula: m = (19 - 4) / (2 - (-3)) m = (15) / (5)

Find the slope

After calculating the value, we find the slope of the line g(x): m = 15 / 5 m = 3 The slope of line g is 3.

Key concepts

Linear Equations

Linear equations are a cornerstone of algebra, representing relationships where one variable depends directly on another through a constant rate of change. Linear equations are typically in the format of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept — the point where the line crosses the y-axis. This simple equation describes a straight line, which can be easily graphed using these two components.

• **Understanding Intercepts**: The y-intercept \(b\) plays a crucial role, indicating where the line starts on the y-axis when \(x = 0\).
• **Parallel Lines**: Lines that are parallel share the same slope \(m\) but have different y-intercepts \(b\).
• **Perpendicular Lines**: If two lines are perpendicular, the product of their slopes is -1, showing how they intersect at right angles.

Linear equations make it easier to relate two changing quantities, making them extremely useful in diverse fields ranging from physics to finance.By knowing how linear equations work, you can predict values, find intersections, and solve numerous real-world problems, where the relationship between variables is straightforward.

Slope Calculation

Slope is an essential concept in algebra that defines the steepness or incline of a line, giving insight into how one variable changes as another changes. It is symbolized by \(m\) in the equation of a line and is calculated as the "rise over run," which translates into the change in \(y\) over the change in \(x\) between any two points on the line.
To calculate the slope:

• Identify two distinct points on the line, commonly labeled as \((x_1, y_1)\) and \((x_2, y_2)\).
• Subtract the y-coordinates and x-coordinates separately to find the differences: \(\Delta y = y_2 - y_1\) and \(\Delta x = x_2 - x_1\).
• Divide the difference in y by the difference in x to derive the slope: \(m = \frac{\Delta y}{\Delta x}\).

The result tells you how much \(y\) will change when \(x\) increases by 1. In the exercise provided, the slope calculation was presented as:\[m = \frac{19 - 4}{2 - (-3)} = \frac{15}{5} = 3\]This confirms the definition of a line and allows you to gauge how steep or flat the line is, which is pivotal for understanding the direction and rate of change between variables.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry to provide a graphical representation of equations through the use of the coordinate plane. This mathematical field allows us to investigate the properties of figures using algebraic equations, thereby bridging the gap between visual and analytical understanding.

• **Points and Lines**: Each point on the plane is defined by a coordinate pair \((x, y)\), and lines are plotted using equations like \(y = mx + b\).
• **Distance and Midpoint**: You can calculate the distance between any two points using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) and find the midpoint using \( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
• **Graphing Linear Equations**: By plotting lines on the coordinate plane, concepts such as slope and intercepts become visually interpretable.

Coordinate geometry effectively translates equations into geometric shapes, allowing for an intuitive understanding of algebraic principles. This powerful method not only aids in directly solving problems but also provides insights into geometric concepts through an algebraic lens. Understanding how coordinates interact within this framework is fundamental for advanced studies in mathematics and related applications.