Problem 87 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. To find the \(y\) -intercept of a graph, let \(x=0\) and solve the equation for \(y\).

Step-by-Step Solution

Answer

The statement is true. For any equation, setting x=0 and solving for y will yield the y-intercept of the graph.

Step 1: Understand the concept of y-intercept

In coordinate geometry, the y-intercept of a line or curve is the point where it crosses the y-axis. This point has an x-coordinates of zero, because it lies on the y-axis, where x is always zero. Therefore, to find the y-intercept of a line or graph, one should let x = 0 and solve for y in the given equation.

Step 2: Evaluation of Statement

Given the explanation in step 1, it can be concluded that the statement 'To find the y -intercept of a graph, let x=0 and solve the equation for y.' is fundamentally correct, as this is the method for finding the y-intercept of an equation or function.

Key Concepts

Graphing equations

To graph an equation, you essentially plot a line or curve that represents all solutions to the equation. This visual representation helps to see the relationship between variables clearly. For linear equations like \( y = mx + b \), the graph will be a straight line. Each point on this line satisfies the equation.

Graphing involves:

• Identifying variables: Typically \( x \) and \( y \) are used in Cartesian coordinates.
• Determining points: Choose values for \( x \) and calculate corresponding \( y \) values to plot.
• Drawing the line or curve: Connect these points to display the entire equation's solutions.

Graphs can take on different shapes depending on the type of equation, such as linear, quadratic, or exponential. Recognizing the type of equation can help you predict the graph's shape and behavior.

Y-intercept

The y-intercept is a crucial concept in coordinate geometry and occurs where the graph of an equation intersects the y-axis. At this point, the value of \( x \) is always zero. This is because the y-axis is vertical, representing the points where \( x = 0 \).

The importance of y-intercepts includes:

• Understanding initial conditions: In real-world applications, the y-intercept often represents starting conditions.
• Charting linear equations: For linear equations \( y = mx + b \), \( b \) is the y-intercept, where the line crosses the y-axis.
• Facilitating graphing: Knowing the y-intercept aids in laying the graph accurately, providing a starting plot point.

To find the y-intercept of an equation, set \( x = 0 \) and solve for \( y \). This method is reliable for any function or equation where a y-intercept exists.

Solving equations

Solving equations involves finding the values of variables that satisfy the given equation. It's essential in graphing, understanding intercepts, and making sense of mathematical relationships.

Key steps in solving equations include:

• Simplifying: Combine like terms and simplify the equation as much as possible.
• Isolating variables: Use techniques such as addition, subtraction, multiplication, and division to isolate the variable of interest.
• Checking solutions: Verify that your solution satisfies the original equation.

Solving for \( y \) by plugging \( x = 0 \) is a specific method used to find the y-intercept in coordinate geometry. By mastering these solving techniques, you can effectively tackle various mathematical challenges and appreciate the interconnected nature of algebraic concepts.