Problem 37 - Algebra and Trigonometry

Question

\(37-40\) An equation and its graph are given. Find the \(x\) - and \(y\) -intercepts. $$ y=4 x-x^{2} $$

Step-by-Step Solution

Answer

The x-intercepts are (0, 0) and (4, 0); the y-intercept is (0, 0).

Step 1: Understand the Problem

We are tasked with finding the x-intercepts and the y-intercept of the equation \(y=4x-x^2\). An intercept is a point where the line or curve crosses the axis: the x-intercept is where it crosses the x-axis, and the y-intercept is where it crosses the y-axis.

Step 2: Finding the y-intercept

To find the y-intercept, we set \(x = 0\) in the equation and solve for \(y\). Substituting into the equation: \[y = 4(0) - (0)^2 = 0\]So the y-intercept is at the point \((0, 0)\).

Step 3: Finding the x-intercepts

To find the x-intercepts, set \(y = 0\) in the equation and solve for \(x\): \[ 0 = 4x - x^2 \]This simplifies to: \[ x^2 - 4x = 0 \]Factor the equation: \[ x(x - 4) = 0 \]Therefore, \(x = 0\) or \(x = 4\). So the x-intercepts are at the points \((0, 0)\) and \((4, 0)\).

Step 4: Conclusion

The y-intercept of the equation is \((0, 0)\) and the x-intercepts are \((0, 0)\) and \((4, 0)\). The point \((0, 0)\) is both an x-intercept and a y-intercept because it is where the graph touches both axes.

Key Concepts

x-intercepts

An x-intercept is where the graph of an equation touches or crosses the x-axis. This represents the point(s) where the value of the function is zero. In simpler words, these are the values of \(x\) for which \(y = 0\).
To find the x-intercepts for any equation, set the equation equal to zero and solve for \(x\).

• In our example, the equation is \(y = 4x - x^2\). We set \(y = 0\) to find the x-intercepts: \[0 = 4x - x^2\]
• Solve this equation to find the x-intercepts by factoring or using the quadratic formula, as needed.
  In our exercise, we factor: \(x(x - 4) = 0\), giving solutions \(x = 0\) and \(x = 4\).

So, the x-intercepts are at the points \((0, 0)\) and \((4, 0)\).

y-intercepts

The y-intercept is the point where the graph crosses the y-axis, which means it's the value of \(y\) when \(x = 0\).
Finding it is straightforward:

• Substitute \(x = 0\) into the equation.
• Calculate the value of \(y\) at this point.
  In our equation \(y = 4x - x^2\), substitute \(x = 0\):
• \[y = 4(0) - (0)^2 = 0\]

So, the y-intercept is at the point \((0, 0)\).
Interestingly, in this case, the y-intercept is also an x-intercept as the graph passes through the origin.

quadratic equation

A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
Characteristics of quadratics:

• The graph of a quadratic equation is a parabola.
• The parabola can open upwards or downwards depending on the sign of \(a\): upwards if \(a > 0\) and downwards if \(a < 0\).
• Quadratic equations can have two, one, or no real solutions.

In our exercise, the equation \(y = 4x - x^2\) is a quadratic equation:

• Rewriting it in the standard form gives \(-x^2 + 4x + 0 = 0\).
• Identifying the coefficients: \(a = -1\), \(b = 4\), \(c = 0\).

This results in a downward opening parabola due to \(a\) being negative.

factoring equations

Factoring is a technique used to solve quadratic equations by expressing them as a product of linear factors.
This step can simplify equations and make finding roots easier. The goal is to set the equation to zero first.
In our exercise, we have the equation \(x^2 - 4x = 0\).
Factoring steps include:

• First, factor out the greatest common factor; here, it is \(x\).
• Rewrite the equation: \(x(x - 4) = 0\).
• From this, we can easily identify the solutions by setting each factor equal to zero:
  • \(x = 0\)
  • \(x - 4 = 0\), leading to \(x = 4\)

Thus, factoring helped us find the roots \(x = 0\) and \(x = 4\), which are the x-intercepts of our quadratic equation.