Graph of Quadratic Polynomial

A quadratic polynomial is a polynomial of degree 2, written in the standard form p(x) = ax^2 + bx + c, where a, b, and c are real numbers and a is not equal to zero. The graph of this polynomial is obtained by plotting the points (x, p(x)) for various values of x on the Cartesian plane and joining them with a smooth curve.

Parabola: The graph of every quadratic polynomial is a curve called a parabola. A parabola is symmetric about a vertical line called the axis of symmetry. The point where the parabola turns (its lowest or highest point) is called the vertex.

Direction of Opening:

• If a > 0 (positive leading coefficient), the parabola opens upward (U-shape). The vertex is the minimum point.
• If a < 0 (negative leading coefficient), the parabola opens downward (inverted U-shape). The vertex is the maximum point.

Vertex: The vertex of the parabola y = ax^2 + bx + c is at the point (-b/(2a), p(-b/(2a))). The x-coordinate of the vertex is -b/(2a), and the y-coordinate is obtained by substituting this value into the polynomial.

Axis of Symmetry: The vertical line x = -b/(2a) passes through the vertex and divides the parabola into two mirror-image halves.

Zeroes from the Graph: The zeroes of the polynomial are the x-coordinates of the points where the parabola intersects the x-axis. A quadratic polynomial can have 0, 1, or 2 real zeroes, depending on whether the parabola does not touch the x-axis, touches it at exactly one point, or crosses it at two points.

The shape of the graph of a quadratic polynomial can be understood by completing the square.

Step 1: Start with y = ax^2 + bx + c.

Step 2: Factor out 'a' from the first two terms: y = a(x^2 + (b/a)x) + c.

Step 3: Complete the square inside the parentheses. To complete the square for x^2 + (b/a)x, add and subtract (b/(2a))^2 = b^2/(4a^2):

y = a[x^2 + (b/a)x + b^2/(4a^2) - b^2/(4a^2)] + c

= a[(x + b/(2a))^2 - b^2/(4a^2)] + c

= a(x + b/(2a))^2 - b^2/(4a) + c

= a(x + b/(2a))^2 + (4ac - b^2)/(4a)

Step 4: This is now in vertex form: y = a(x - h)^2 + k, where h = -b/(2a) and k = (4ac - b^2)/(4a).

Why this gives a parabola: The expression a(x - h)^2 + k shows that y is a squared function of (x - h), shifted by h horizontally and k vertically. The squared term (x - h)^2 is always non-negative, so:

If a > 0: y >= k for all x, with minimum value k at x = h. The parabola opens upward.

If a < 0: y <= k for all x, with maximum value k at x = h. The parabola opens downward.

Why the parabola is symmetric: Notice that (x - h)^2 = (-(x - h))^2. This means points equidistant from x = h on either side give the same y-value. Hence the parabola is symmetric about the line x = h.

Effect of the coefficient 'a': The magnitude of 'a' determines how wide or narrow the parabola is. A larger |a| makes the parabola narrower (steeper), while a smaller |a| makes it wider (flatter). The sign of 'a' determines the direction of opening.

Connection to zeroes: Setting y = 0 gives a(x - h)^2 + k = 0, or (x - h)^2 = -k/a. This has real solutions only when -k/a >= 0. Substituting k = (4ac - b^2)/(4a) and simplifying leads to the condition b^2 - 4ac >= 0, which is the discriminant condition.

The graph of a quadratic polynomial can take several forms depending on the coefficients and the discriminant:

Type 1: Upward-opening parabola with two x-intercepts (a > 0, D > 0)

Example: y = x^2 - 5x + 6. Here a = 1 > 0 and D = 25 - 24 = 1 > 0. The graph opens upward and crosses the x-axis at x = 2 and x = 3. The vertex is below the x-axis.

Type 2: Upward-opening parabola with one x-intercept (a > 0, D = 0)

Example: y = x^2 - 4x + 4 = (x - 2)^2. The graph touches the x-axis at x = 2 (the vertex lies on the x-axis). Both zeroes are equal.

Type 3: Upward-opening parabola with no x-intercept (a > 0, D < 0)

Example: y = x^2 + x + 1. Here D = 1 - 4 = -3 < 0. The entire graph lies above the x-axis. There are no real zeroes.

Type 4: Downward-opening parabola with two x-intercepts (a < 0, D > 0)

Example: y = -x^2 + 4x - 3. Here a = -1 < 0 and D = 16 - 12 = 4 > 0. The graph opens downward and crosses the x-axis at x = 1 and x = 3. The vertex is above the x-axis.

Type 5: Downward-opening parabola with one x-intercept (a < 0, D = 0)

Example: y = -x^2 + 6x - 9 = -(x - 3)^2. The vertex is on the x-axis at x = 3, and the parabola opens downward.

Type 6: Downward-opening parabola with no x-intercept (a < 0, D < 0)

Example: y = -x^2 - 2x - 5. Here D = 4 - 20 = -16 < 0. The entire graph lies below the x-axis. No real zeroes.

Method: Steps to Sketch the Graph of a Quadratic Polynomial

Step 1: Identify a, b, and c from the polynomial y = ax^2 + bx + c.

Step 2: Determine the direction of opening. If a > 0, the parabola opens upward. If a < 0, it opens downward.

Step 3: Find the vertex. The x-coordinate of the vertex is h = -b/(2a). Substitute h into the polynomial to find the y-coordinate: k = p(h).

Step 4: Find the y-intercept by setting x = 0: y-intercept = (0, c).

Step 5: Find the x-intercepts (zeroes) by solving ax^2 + bx + c = 0. Calculate D = b^2 - 4ac. If D >= 0, the zeroes are x = (-b + sqrt(D))/(2a) and x = (-b - sqrt(D))/(2a).

Step 6: Plot the vertex, y-intercept, x-intercepts (if any), and a few additional points on either side of the vertex.

Step 7: Draw a smooth parabolic curve through these points, ensuring symmetry about the axis of symmetry x = -b/(2a).

Reading Zeroes from a Graph: To find the zeroes from a given graph, identify the x-coordinates where the curve crosses or touches the x-axis. These x-values are the zeroes of the polynomial.

The graph of a quadratic polynomial has numerous applications in mathematics, science, and everyday life.

Projectile Motion: When an object is thrown upward at an angle, its path (trajectory) is a parabola. The quadratic polynomial models the height as a function of time or horizontal distance. The maximum height corresponds to the vertex, and the zeroes represent the launch and landing points.

Optimisation Problems: Since the vertex of a parabola gives either the maximum or minimum value, quadratic functions are used to solve optimisation problems. For example, finding the maximum area that can be enclosed with a fixed perimeter, or the price that maximises revenue.

Architecture and Engineering: Parabolic shapes appear in bridges, satellite dishes, and headlight reflectors. The reflective property of parabolas (all rays parallel to the axis reflect through the focus) makes them ideal for these applications.

Economics: Revenue and profit functions are often quadratic. The graph helps businesses identify break-even points (zeroes) and maximum profit (vertex).

Sports: The trajectory of a ball in cricket, basketball, or football follows a parabolic path, and understanding the graph helps in analysing and predicting the ball's motion.