Question

Which of the following equations has a graph for which all roots are greater than \(0 ?\) a. \(y=4|x|\) b. \(y=x^{2}-4\) c. \(y=(x-2)^{2}\) d. \(y=x(x-2)^{2}\)

Short answer

The correct answer is c. \(y=(x-2)^{2}\).

Step-by-step solution

Analyze Option a

For the equation \(y=4|x|\), the graph is a V-shape that touches the origin. The roots are at \(x=0\). This means it does not satisfy the condition of having all roots greater than 0.

Analyze Option b

For the equation \(y=x^{2}-4\), the graph is a parabola that opens upwards with roots where \(x=2\) and \(x=-2\). Since one of the roots is negative, this option does not satisfy the condition.

Analyze Option c

For the equation \(y=(x-2)^{2}\), the graph is a parabola that opens upwards shifted right by 2 units, and the only root is at \(x=2\). Since the root is greater than 0, this option satisfies the condition.

Analyze Option d

For the equation \(y=x(x-2)^{2}\), set \(y=0\) to find the roots. Solving \(x(x-2)^{2}=0\) gives roots at \(x=0\) and \(x=2\). Since one of the roots is 0, this option does not satisfy the condition.

Conclusion

Based on the analysis, only the equation \(y=(x-2)^{2}\) has all roots greater than 0.

Key concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\). They represent parabolas when graphed on the coordinate plane. These equations have various properties and features, such as:

• **The Vertex:** The highest or lowest point on the graph of the quadratic equation.
• **Roots or Zeros:** The values of \(x\) where the equation equals zero.
• **Symmetry:** The graph of a quadratic equation is symmetric about a vertical line called the axis of symmetry.

To solve these equations, you can use methods like factoring, completing the square, or the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). These methods help in finding the roots, which are essential for graph analysis.

Graph Analysis

Analyzing the graph of a quadratic equation is key to understanding its properties. Here are some important steps to consider:

• **Identify the Vertex:** This is the point \((h, k)\), where the graph changes direction.
• **Determine the Direction:** If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards.
• **Find the Roots:** These are the points where the graph intersects the x-axis.

In the given exercise, analyzing the different equations involves looking at how the graph interacts with the x-axis. For example, in option c, \(y=(x-2)^2\), the graph is a parabola that opens upwards, and its root at \(x=2\) is positive, meeting the required condition.

Roots of Equations

Roots, also known as zeros or solutions, are the values of \(x\) that satisfy the equation \(f(x) = 0\). They are crucial in understanding the behavior of a graph.

• **Real Roots:** If the quadratic equation intersects the x-axis at real numbers, these intersection points are the real roots.
• **Complex Roots:** If the quadratic does not intersect the x-axis, the roots are complex and involve imaginary numbers.

In our exercise, the equation \(y=(x-2)^2\) has a root at \(x=2\) which is greater than 0. This makes option c the correct choice where all roots are positive. Analyzing roots helps in predicting graph behaviour and solving real-world problems efficiently.