Problem 53 - Algebra 1

Question

Does the graph of the function contain the point \((0,1) ?\) $$y=-3^{x}$$

Step-by-Step Solution

Answer

No, the point \((0,1)\) does not lie on the graph of the function \(y=-3^{x}\).

Step 1: Understand the function

The function in question is \(y=-3^{x}\). This function returns the negative of three raised to the power of x.

Step 2: Substitute the point into the function

The x and y coordinates of the point are 0 and 1, respectively. Therefore, by substituting these values into the function,y becomes 1 and x becomes 0. So, after substitution, the equation becomes 1 = -3^{0}.

Step 3: Evaluate the equation

The expression -3^{0} equals -1 because any number (except zero) to the power of 0 is 1, but since it's -3^{0} the result will be -1.

Step 4: Compare the results

After evaluating, the result is -1. But y coordinate of given point is 1. Hence, the point \((0,1)\) does not satisfy the function.

Key Concepts

Function Evaluation

Evaluating a function means determining the output for a given input. In practice, we are taking the input value, plugging it into the function, and calculating the result.
For the function provided here, which is written as \(y = -3^{x}\), the process is straightforward:

• Identify the "input," which is the x-value of the point of interest, in our case \(x = 0\).
• Substitute this into the function, replacing \(x\) with 0.
• Calculate \(-3^{0}\) to determine the value of \(y\).

By following these steps, you'll be able to evaluate the function at any given x-value and find the corresponding y-value, which is essential for graphing functions.

Graph Points

When working with functions, graph points refer to specific locations on a graph defined by coordinates. A point such as \((0, 1)\) tells us exactly where to look on a graph: where the x-coordinate is 0 and the y-coordinate is 1.
To check if any point lies on the graph of a function, like \(y = -3^{x}\), you substitute x into the equation and verify if the resulting y is the same as the y-coordinate of the point. If so, the point is on the graph. Otherwise, it is not.
This method is useful not only in verifying where points lie in relation to a graph but also in sketching the graph itself, as you'll plot points one-by-one to visualize the function.

Exponential Functions

Exponential functions are intriguing since their variables are in the exponent position, for example, \(y = -3^x\). Understanding this type of function is crucial because:

• They show rapid growth or decay, evident when graphing them.
• The base (here it's 3) determines the growth rate.
• Negative signs before the expression like \(-3^{x}\) indicate the function's output will always be negative, flipping the graph vertically.

Because of this exponential form, small changes in the x-value can result in large changes in y-values, making these functions fascinating and powerful in modeling real-world phenomena such as population growth or radioactive decay.

Coordinate Substitution

Coordinate substitution involves replacing the variables in an equation with specific numbers from a coordinate point.
In our example, we substitute the point \((0, 1)\) into the function \(y = -3^{x}\), replacing \(x\) with 0 to see if we get 1 for \(y\). This verification proceeds in a few steps:

• Input the x-coordinate into the function.
• Compute the output and compare it with the y-coordinate.

Coordinate substitution is pivotal in verifying if a point lies on a given function or not. It's a practical way to engage with functions analytically and visually, using graphs to confirm your calculations and vice versa.