Problem 55 - Algebra 1

Question

Does the graph of the function contain the point \((0,1) ?\) $$y=4 \cdot 1^{x}$$

Step-by-Step Solution

Answer

No, the point (0,1) does not lie on the graph of the function \(y=4 \cdot 1^{x}\).

Step 1: Substitute the x-coordinate

Substitute the x-coordinate of the point into the function. The given x-coordinate is 0, so \(y=4 \cdot 1^{0}\).

Step 2: Simplify the expression

Since any real number raised to the power of 0 is 1, the expression simplifies to \(y=4 \cdot 1=4\).

Step 3: Compare the y-coordinate

Compare the resulting y-coordinate with the y-coordinate of the given point. If they match, the point lies on the graph. Here, the resulting y-coordinate is 4, while the y-coordinate of the point given is 1. They do not match.

Key Concepts

Understanding the Substitution Method

When you work with function graphing, one essential technique is the substitution method. This involves replacing a variable with a specific number to see how it affects the entire expression or equation. In our original exercise, we substitute the x-coordinate from the point into the function equation.

• Find the equation of the function. In this case, it's given as \(y=4 \cdot 1^x\).
• Substitute \(x = 0\) into the function equation for \(x\), so the equation becomes \(y = 4 \cdot 1^0\).
• Evaluate the result from this substitution to find the y-coordinate associated with the x-value.

This method is particularly useful because it reduces the equation to a simpler form, allowing for straightforward calculations. By substituting numbers for variables, we can test specific points to see whether they lie on the graph of a function.

Exploring Powers of Zero

The concept of a number raised to the power of zero is a fundamental rule in mathematics: any non-zero number raised to the power of zero equals one. Understanding this concept is crucial in simplifying expressions and solving equations.

• The expression \(1^0\) simplifies to 1 because of this rule.
• No matter what the base is, as long as it is not zero, any number to the power of zero will always be 1.
• For instance, whether the base is 10, 2, or 0.5, the result of raising it to the zero power remains 1.

This simplifies calculations significantly, such as in our step-by-step solution where \(4 \cdot 1^0\) simplifies to \(4 \cdot 1\), which further simplifies to 4. Remembering this rule helps solve many similar mathematical problems with ease.

Expression Simplification Techniques

Simplifying expressions is a cornerstone of function graphing and algebra in general. Once you've performed substitution or have expressions that contain powers of zero, it's often necessary to simplify these expressions to find easy-to-understand results.

• Simplifying means reducing the expression to its most concise and manageable form.
• In our example, \(4 \cdot 1\) is already in its simplest form, equal to 4.
• Simplification involves using arithmetic operations like multiplication or addition and applying exponent rules, such as the powers of zero, to achieve a straightforward result.

By breaking down complex expressions into their simplest forms, you not only make calculations and comparisons easier, but you also gain clearer insight into the behavior of functions and the relationships between their variables.