Problem 44 - Precalculus

Question

Show that the slope of the line passing through the two points (3,9) and \(\left(3+h,(3+h)^{2}\right)\) is \(6+h\).

Step-by-Step Solution

Answer

The slope of the line is \(6 + h\).

Step 1: Understand the Formula for Slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In this problem, \((x_1, y_1) = (3, 9)\) and \((x_2, y_2) = (3+h, (3+h)^2)\).

Step 2: Substitute the Coordinates into the Slope Formula

Substitute the given points into the slope formula:\[ m = \frac{(3+h)^2 - 9}{(3+h) - 3} \]This gives us the expression to simplify in the next step.

Step 3: Simplify the Expression for the Slope

First, expand \((3+h)^2\) as follows: \((3+h)^2 = 9 + 6h + h^2\). Substitute back into the expression:\[ m = \frac{9 + 6h + h^2 - 9}{h} \]Cancel out the \(9\) from the numerator:\[ m = \frac{6h + h^2}{h} \]

Step 4: Factor out \(h\) from the Numerator

Factor \(h\) out in the numerator:\[ m = \frac{h(6 + h)}{h} \]Since \(h eq 0\), we can cancel \(h\) from the numerator and the denominator.

Step 5: Write the Final Expression for the Slope

After cancellation, the expression for the slope simplifies to:\[ m = 6 + h \]This confirms that the slope of the line passing through the given points is indeed \(6 + h\).

Key Concepts

coordinate geometry

Coordinate geometry allows us to represent geometric figures in a numerical way, using the Cartesian coordinate system. In this system, each point on a plane is represented by a pair of numbers, known as coordinates.
One important aspect of coordinate geometry is finding the distance between points, slopes of lines, and equations of those lines. The slope is a measure of how steep a line is, and it is calculated using the differences in the y-coordinates and x-coordinates of two points.

• Coordinate Points: These are objectives of two numbers representing the x and y values. For example, (3, 9) means the point is three units right and nine units up from the origin.
• Slope Calculation: The formula to find the slope, which we call 'm', is crucial here. It is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

When you have a changing coordinate, like \(3 + h\), you still apply the same principles, adjusting the formulae accordingly. The ability to analyze and find slopes from coordinates makes solving problems with linear equations easier.

linear equations

Linear equations represent straight lines and are fundamental in mathematics. These equations are usually in the form \(y = mx + c\), where 'm' is the slope, and 'c' is the y-intercept.
Understanding linear equations involves recognizing the relationship between the two variables. When graphed on a coordinate plane, these equations always form a straight line. For example, by using the slope formula for the points given in our problem, we established a new equation: \(m = 6 + h\), showing that the slope changes depending on 'h'.

• Standard Form: A common way of expressing linear equations: \(Ax + By = C\).
• Slope-Intercept Form: Useful for understanding how fast a function grows or shrinks: \(y = mx + c\).

This concept is often applied to make predictions, find trends, or solve real-life problems, making understanding it not only helpful for math problems but also practical applications.

problem solving in mathematics

Problem solving is at the heart of mathematics. It involves understanding the problem, devising a plan, carrying out the plan, and then checking the solution for errors.
The exercise shown exemplifies these stages.

• Understanding: begins with comprehending the problem statement and the data provided. Here, we are asked to find the slope.
• Devising a Plan: involves applying the formula for the slope, \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Knowing the formula simplifies the approach.
• Executing: Implementing the plan through calculations and simplifications, like expanding \((3+h)^2\).
• Review: Finally, verifying the results are consistent with what was asked, ensuring our final slope \(6 + h\) is valid and correctly derived.

By approaching problems in this structured way, complex questions can become simple and manageable. Simplicity, accuracy, and clarity are key in mathematical problem solving.