Draw The Graph Of The Given Polynomial P(x) = X² - 6x + 9 And Find Its Zeroes. Provide Justification For Your Answer.

Jun 19, 2025 by ADMIN 118 views

Drawing Polynomial Graphs and Finding Zeroes for p(x) = x² - 6x + 9

Introduction

In this article, we will delve into the process of drawing the graph of the polynomial p(x) = x² - 6x + 9 and finding its zeroes. Understanding how to graph polynomials and identify their zeroes is a fundamental concept in algebra, providing valuable insights into the behavior and properties of polynomial functions. This exercise not only enhances our graphical interpretation skills but also reinforces our understanding of the relationship between algebraic expressions and their geometric representations. We will explore the steps involved in sketching the graph, including identifying key features such as the vertex and axis of symmetry, and then proceed to determine the zeroes of the polynomial, which are the points where the graph intersects the x-axis. Furthermore, we will justify our findings using both graphical analysis and algebraic methods, ensuring a comprehensive understanding of the solution. This comprehensive approach will help solidify your understanding of quadratic functions and their graphical representations, providing a strong foundation for more advanced mathematical concepts.

Understanding the Polynomial

Our primary focus is on the quadratic polynomial p(x) = x² - 6x + 9. This is a polynomial of degree 2, which means its graph will be a parabola. Quadratic polynomials are characterized by their U-shaped curves, which can either open upwards or downwards depending on the coefficient of the x² term. In this case, the coefficient of x² is 1, which is positive, indicating that the parabola will open upwards. The general form of a quadratic polynomial is ax² + bx + c, where a, b, and c are constants. In our polynomial, a = 1, b = -6, and c = 9. These coefficients play a crucial role in determining the shape and position of the parabola on the coordinate plane. The vertex, which is the lowest point on the parabola when it opens upwards, is a key feature we will need to identify. The axis of symmetry, a vertical line that passes through the vertex, divides the parabola into two symmetrical halves. Understanding these characteristics is essential for accurately sketching the graph of the polynomial and finding its zeroes. By analyzing the coefficients and the form of the polynomial, we can predict the basic shape and orientation of the graph, making the graphing process more intuitive and efficient. This initial analysis sets the stage for a more detailed exploration of the polynomial's behavior and its graphical representation.

Finding the Vertex

The vertex of a parabola is a crucial point for graphing a quadratic polynomial. For the polynomial p(x) = x² - 6x + 9, the vertex represents the minimum point of the parabola since the coefficient of x² is positive. To find the vertex, we can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic polynomial in the form ax² + bx + c. In our case, a = 1 and b = -6. Plugging these values into the formula, we get x = -(-6) / (2 * 1) = 6 / 2 = 3. This gives us the x-coordinate of the vertex. To find the y-coordinate, we substitute x = 3 back into the polynomial: p(3) = (3)² - 6(3) + 9 = 9 - 18 + 9 = 0. Therefore, the vertex of the parabola is at the point (3, 0). This point is not only the minimum value of the function but also lies on the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. In this case, the axis of symmetry is the line x = 3. Knowing the vertex and the axis of symmetry provides a solid foundation for sketching the graph, as we have a reference point and a line of symmetry to guide our drawing. The vertex is particularly important because it helps us understand the range of the function and the overall shape of the parabola. With the vertex determined, we can proceed to find other points on the graph to create a more accurate representation of the polynomial function.

Determining the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. For the polynomial p(x) = x² - 6x + 9, we have already found the x-coordinate of the vertex to be x = 3. The axis of symmetry is simply the vertical line defined by this x-coordinate. Therefore, the equation of the axis of symmetry is x = 3. This line serves as a central reference point when graphing the parabola, as any point on one side of the axis will have a corresponding point on the other side at the same vertical distance from the vertex. The axis of symmetry is crucial for understanding the symmetry of the quadratic function and for efficiently plotting points on the graph. For instance, if we know a point on the parabola at x = 2, we can easily find the corresponding point on the other side of the axis of symmetry at x = 4. This symmetry significantly reduces the number of calculations needed to sketch the graph accurately. The axis of symmetry also provides insight into the behavior of the function; it indicates the line around which the function's values are mirrored. In the case of p(x) = x² - 6x + 9, the axis of symmetry at x = 3 confirms that the function is symmetrical around this line, which is a characteristic feature of quadratic functions. With the axis of symmetry clearly defined, we can use it as a guide to plot additional points and complete the graph of the polynomial.

Finding Additional Points

To accurately sketch the graph of p(x) = x² - 6x + 9, we need to find additional points besides the vertex. We can do this by choosing several x-values and calculating the corresponding y-values using the polynomial equation. It's a good practice to select x-values on both sides of the axis of symmetry (x = 3) to ensure a balanced representation of the parabola. Let's choose x = 1, x = 2, x = 4, and x = 5.

For x = 1: p(1) = (1)² - 6(1) + 9 = 1 - 6 + 9 = 4. So, we have the point (1, 4).
For x = 2: p(2) = (2)² - 6(2) + 9 = 4 - 12 + 9 = 1. So, we have the point (2, 1).
For x = 4: p(4) = (4)² - 6(4) + 9 = 16 - 24 + 9 = 1. So, we have the point (4, 1). Notice that this point is symmetrical to (2, 1) with respect to the axis of symmetry.
For x = 5: p(5) = (5)² - 6(5) + 9 = 25 - 30 + 9 = 4. So, we have the point (5, 4). This point is symmetrical to (1, 4).

These additional points, along with the vertex (3, 0), give us a clear picture of the parabola's shape. By plotting these points on a coordinate plane and connecting them with a smooth curve, we can create an accurate graph of the polynomial. The symmetry we observed in the y-values for points equidistant from the axis of symmetry reinforces the symmetrical nature of the parabola. Finding additional points is crucial for capturing the curvature and overall behavior of the polynomial function. With a sufficient number of points plotted, we can confidently sketch the graph and analyze its features, including its zeroes and minimum value. This step ensures that our graphical representation accurately reflects the polynomial function.

Sketching the Graph

With the vertex at (3, 0), the axis of symmetry at x = 3, and the additional points (1, 4), (2, 1), (4, 1), and (5, 4), we can now sketch the graph of the polynomial p(x) = x² - 6x + 9. Start by plotting the vertex, which is the lowest point on the graph. Then, draw the axis of symmetry as a vertical dashed line through x = 3. Plot the additional points we calculated, ensuring they are symmetrical with respect to the axis of symmetry. Now, connect the points with a smooth, U-shaped curve to form the parabola. The parabola opens upwards because the coefficient of x² is positive. The graph should be symmetrical around the axis of symmetry, with the vertex as the minimum point. The shape of the parabola will become clearer as you connect the points, creating a visual representation of the polynomial function. A well-sketched graph allows us to easily identify the key features of the polynomial, such as its vertex, axis of symmetry, and zeroes. It also provides a visual confirmation of the algebraic properties we have calculated. Sketching the graph is not just a mechanical process; it's an integral part of understanding the behavior of the polynomial function. It allows us to see the relationship between the algebraic expression and its geometric representation, reinforcing our understanding of quadratic functions. With the graph sketched, we can now proceed to find the zeroes of the polynomial, which are the points where the graph intersects the x-axis.

Finding the Zeroes Graphically

The zeroes of a polynomial are the points where the graph intersects the x-axis, which means p(x) = 0. By examining the graph of p(x) = x² - 6x + 9, we can visually identify these points. In our case, the parabola touches the x-axis at only one point, which is the vertex (3, 0). This indicates that the polynomial has one real zero, which is x = 3. Graphically, the zeroes are the x-coordinates of the points where the curve crosses or touches the x-axis. If the parabola did not intersect the x-axis at all, it would mean that the polynomial has no real zeroes. If it intersected at two distinct points, it would have two real zeroes. The number of zeroes and their values provide valuable information about the polynomial's behavior and its solutions. Finding the zeroes graphically is a straightforward process once the graph is accurately sketched. It allows us to quickly identify the solutions to the equation p(x) = 0 without resorting to algebraic methods. However, it's important to verify the graphical solution algebraically to ensure accuracy and completeness. The graphical method provides a visual confirmation of the zeroes, which can be particularly helpful in understanding the nature of the roots of the polynomial. In this case, the graph clearly shows that the polynomial has a single real zero at x = 3, which we will confirm algebraically in the next section.

Finding the Zeroes Algebraically

To find the zeroes algebraically, we need to solve the equation p(x) = x² - 6x + 9 = 0. This is a quadratic equation, and we can solve it by factoring, completing the square, or using the quadratic formula. In this case, the polynomial is a perfect square trinomial, which makes factoring the easiest method. We can rewrite the equation as (x - 3)² = 0. Taking the square root of both sides, we get x - 3 = 0, which simplifies to x = 3. This confirms that the polynomial has one real zero at x = 3. The fact that we obtained a single solution algebraically aligns perfectly with our graphical observation that the parabola touches the x-axis at only one point. Algebraic methods provide a precise way to find the zeroes of a polynomial, especially when the graph may not give a clear reading due to scaling or inaccuracies in sketching. Factoring is a powerful technique when the polynomial can be expressed as a product of simpler factors. However, for more complex quadratic equations, the quadratic formula can be used to find the zeroes. The algebraic solution not only confirms the graphical solution but also provides a deeper understanding of the polynomial's roots. In this case, the single zero at x = 3 indicates that the graph touches the x-axis at the vertex, which is a characteristic feature of perfect square trinomials. The combination of graphical and algebraic methods provides a robust approach to finding and verifying the zeroes of a polynomial, ensuring a comprehensive understanding of its behavior.

Justifying the Answer

To justify our answer, we need to show that the zeroes we found graphically and algebraically are consistent and accurate. We found that the graph of p(x) = x² - 6x + 9 touches the x-axis at (3, 0), indicating a zero at x = 3. Algebraically, we solved the equation x² - 6x + 9 = 0 and also found the zero to be x = 3. This consistency between the graphical and algebraic methods provides strong justification for our answer. The fact that both approaches yield the same result reinforces our confidence in the solution. Furthermore, we can verify our solution by substituting x = 3 back into the original polynomial: p(3) = (3)² - 6(3) + 9 = 9 - 18 + 9 = 0. This confirms that x = 3 is indeed a zero of the polynomial. Justification is a crucial step in mathematical problem-solving. It ensures that our solution is not only correct but also well-supported by evidence. By using multiple methods and verifying the results, we can strengthen our understanding of the concepts and the problem-solving process. In this case, the agreement between the graphical and algebraic solutions, along with the substitution check, provides a comprehensive justification for our answer. This approach not only solves the problem but also enhances our ability to reason mathematically and communicate our findings effectively. The thorough justification process underscores the importance of precision and accuracy in mathematical analysis.

Conclusion

In conclusion, we have successfully drawn the graph of the polynomial p(x) = x² - 6x + 9 and found its zeroes using both graphical and algebraic methods. We identified the vertex as (3, 0) and the axis of symmetry as x = 3. By plotting additional points, we sketched the parabola, which opens upwards and touches the x-axis at only one point. Graphically, we determined that the polynomial has a single zero at x = 3. Algebraically, we factored the polynomial and solved the equation x² - 6x + 9 = 0, confirming the zero at x = 3. The consistency between the graphical and algebraic solutions, along with the substitution check, provides strong justification for our answer. This exercise has demonstrated the relationship between algebraic expressions and their geometric representations. It has also highlighted the importance of using multiple methods to solve and verify mathematical problems. Understanding how to graph polynomials and find their zeroes is a fundamental skill in algebra, with applications in various fields of mathematics and beyond. The comprehensive approach we have taken, from understanding the polynomial to justifying the answer, ensures a deep and thorough understanding of the concepts involved. This knowledge provides a solid foundation for tackling more complex problems and further exploring the world of polynomial functions. The ability to analyze and interpret polynomial graphs is a valuable asset in mathematical problem-solving and analysis.