Problem 53 - Precalculus Mathematics for Calculus

Question

An equation and its graph are given. Find the x- and y-intercepts. $$y=4 x-x^{2}$$ (graph can't copy)

Step-by-Step Solution

Answer

The x- and y-intercepts are (0, 0) and (4, 0).

Step 1: Finding the x-intercepts

To find the x-intercepts of the equation, we set the y-value to zero because at the x-intercepts, the graph of the equation crosses the x-axis where \(y = 0\). Thus we solve:\[0 = 4x - x^2\]This simplifies to: \[x^2 - 4x = 0\]Factor the equation:\[x(x - 4) = 0\]Setting each factor equal to zero gives the solutions: \(x = 0\) and \(x = 4\). Thus, the x-intercepts are at the points \((0, 0)\) and \((4, 0)\).

Step 2: Finding the y-intercept

The y-intercepts occur where the graph intersects the y-axis, which is where \(x = 0\). Substitute \(x = 0\) in the given equation:\[y = 4(0) - (0)^2 = 0\]Therefore, the y-intercept is at the point \((0, 0)\).

Key Concepts

x-intercepts

To find the x-intercepts of an equation, we need to determine where the graph of the equation crosses the x-axis. The key characteristic of these points is that they have a y-value of zero. Therefore, the general method involves setting the equation equal to zero and solving for x. This will give us the x-values where the graph touches or crosses the x-axis.

In the case of the equation \(y = 4x - x^2\), we set \(y = 0\) to find these points:

• Write the equation as \(0 = 4x - x^2\)
• Rearrange it to \(x^2 - 4x = 0\)
• Factor the equation into \(x(x - 4) = 0\)

Once factored, set each factor equal to zero:

• From \(x = 0\), we find one x-intercept at the point \((0, 0)\).
• From \(x - 4 = 0\), we find another intercept at \((4, 0)\).

Thus, the x-intercepts are at the points \((0, 0)\) and \((4, 0)\). It's important to note that these points provide valuable insights into the behavior of the quadratic function.

y-intercepts

Unlike x-intercepts, y-intercepts are points where the graph of a function crosses the y-axis. At these points, the x-value is always zero because it's where the line meets the vertical axis. This process is much simpler, requiring just a direct substitution of \(x = 0\) into the equation.

For our equation \(y = 4x - x^2\), we can find the y-intercept by substituting \(x = 0\):

• Calculate: \(y = 4(0) - (0)^2\)
• Simplify to find \(y = 0\)

The y-intercept, therefore, is the point \((0, 0)\).

This point tells us that at the origin, the curve of the function intersects the y-axis. Understanding y-intercepts helps us in assessing the starting value or the initial position of the graph, particularly in real-world contexts.

factoring equations

Factoring is a crucial algebraic technique used to solve equations effectively, especially quadratics, by expressing them as a product of simpler expressions. This allows us to set each factor equal to zero, thus making solving these equations straightforward.

Let's consider the equation \(0 = 4x - x^2\), which we rearranged to \(x^2 - 4x = 0\).

To factor this equation, follow these steps:

• Look for common factors in each term. In this case, the common factor is \(x\).
• Factor out \(x\): \(x(x - 4) = 0\)

After factorization, we can solve for the x-values by setting each factor to zero:

• If \(x = 0\), this gives one solution.
• For \(x - 4 = 0\), solve to get \(x = 4\).

Factoring is not just about decorating equations. It's a skillful way to break down complex problems into simpler parts, providing clarity and facilitating calculations. This process reveals critical points such as intercepts which are essential in understanding the graphs of quadratic functions.