Problem 67 - Algebra Form and Function

Question

Which equation, (a)-(d), has the graph that crosses the \(y\) -axis at the highest point? (a) \(y=3(x-1)+5\) (b) \(x=3 y+2\) (c) \(y=1-6 x\) (d) \(2 y=3 x+1\)

Step-by-Step Solution

Answer

Answer: (c) \(y=1-6x\)

Step 1: Rewrite the equations in slope-intercept form

First, we'll rewrite each equation in slope-intercept form (y = mx + b): (a) \(y=3(x-1)+5\): The given equation is already in slope-intercept form. (b) \(x=3 y+2\): We need to solve for y in terms of x. (c) \(y=1-6x\): The given equation is already in slope-intercept form. (d) \(2y=3x+1\): We need to solve for y in terms of x.

Step 2: Solve the equations for y

For equations (b) and (d), we'll isolate y: (b) \(x=3y+2\) Subtract 2 from both sides: \(x-2=3y\) Divide by 3: \(y=\frac{1}{3}x-\frac{2}{3}\) (d) \(2y=3x+1\) Divide by 2: \(y=\frac{3}{2}x+\frac{1}{2}\)

Step 3: Determine the y-intercepts of each equation

Now we will find the y-intercepts of each equation: (a) \(y=3(x-1)+5\): \(b=3(-1)+5=3-5=-2\) (b) \(y=\frac{1}{3}x-\frac{2}{3}\): \(b=-\frac{2}{3}\) (c) \(y=1-6x\): \(b=1\) (d) \(y=\frac{3}{2}x+\frac{1}{2}\): \(b=\frac{1}{2}\)

Step 4: Compare the y-intercepts

Now that we have the y-intercepts for each equation, let's compare them: - Equation (a) has a y-intercept of -2. - Equation (b) has a y-intercept of \(\frac{-2}{3}\). - Equation (c) has a y-intercept of 1. - Equation (d) has a y-intercept of \(\frac{1}{2}\). Among the four equations, equation (c) has the highest y-intercept, with a value of 1.

Step 5: Determine the final answer

Since equation (c) has the highest y-intercept, the graph that crosses the y-axis at the highest point is: (c) \(y=1-6x\)

Key Concepts

y-intercept

In any linear equation written in the form of \(y = mx + b\), the 'b' represents the y-intercept. The y-intercept is the point where the graph of the equation crosses the y-axis. This means that at the y-intercept, the value of \(x\) is zero. Here are a few important aspects of y-intercepts:

• In the equation \(y = mx + b\), simply look at the coefficient 'b' to determine the y-intercept.
• The y-intercept can either be a positive or negative value, which affects where the graph crosses the y-axis.
• Understanding the y-intercept gives you a starting point when you are graphing an equation, making it easier to plot the line on a graph.

In the provided exercise, we observed that each linear equation was transformed into slope-intercept form to identify each equation's y-intercept. The highest y-intercept among the given equations was found to be for equation (c), which is at 1.

Linear Equations

Linear equations are equations of the first order that form a straight line when graphed on a coordinate plane. They can be easily recognized by their slope-intercept formulation: \(y = mx + b\), where:

• \(m\) represents the slope of the line.
• \(b\) is the y-intercept of the line.
• The variables \(x\) and \(y\) are raised to the power of one, making them the simplest form of algebraic equations.

From the exercise, we learned how to rearrange various equations into the familiar form of \(y = mx + b\). Being able to identify and convert different forms of equations into this standard slope-intercept form allows us to easily understand their graphed visualizations and main characteristics like slope and y-intercept. This skill is critical for solving many practical problems in algebra.

Graphing Linear Functions

Graphing linear functions involves plotting points and drawing the line that represents the equation of the function. Each linear function has a constant rate of change, which makes the graph a straight line.Here are the steps to graphing a linear function:

• Identify the y-intercept from the equation, \(b\), and plot it on the y-axis.
• Determine the slope \(m\) which tells you how steep the line is and in which direction it moves. A positive slope means the line goes up, and a negative slope means the line goes down as you move from left to right.
• Use the slope to find another point on the line. Slope is calculated as rise over run, \(m = \frac{\Delta y}{\Delta x}\).
• Draw a line through the points you've plotted, extending it across the graph.

By understanding these steps, you can graph any linear function. In our exercise, comparing the y-intercepts helped us decide which line would cross the y-axis the highest, showcasing the practical application of these graphing skills.