Examples of Quadratic functions

Based on the provided sources, a quadratic function is described by an equation of the form f(x) = ax² + bx + c, where ‘a’ is not equal to 0. The name “quadratic” is related to the term “square”. The graph of any quadratic function is always a parabola.

Here are some examples of quadratic functions and how they are discussed in the sources:

• y = x²:
  • This is given as a standard prototype example.
  • It is the form where b=0 and c=0, and a=1.
  • Points like (0,0), (1,1), (-1,1), (2,4), and (-2,4) can be plotted to graph it.
  • The graph forms an upward parabola shape.
  • For this function, the slope at any point x is 2x. Setting the slope to 0 (2x = 0) gives x=0, which is the x-coordinate of the vertex where the minimum value is attained.
  • The y-coordinate of the vertex (at x=0) is 0, which is the minimum value.
  • It shows symmetry about the y-axis because, for instance, 2² is the same as -2².
• f(x) = x² + 2x + 1:
  • This function can be graphed by generating a table of ordered pairs and plotting them. Examples of points given are (-2,1), (-1,0), (0,1), and (1,4).
  • The axis of symmetry for this function is x = -1. This is found using the formula x = -b/(2a), where a=1 and b=2.
  • The point where the axis of symmetry meets the parabola is the vertex. For this function, the vertex is at x = -1. Substituting x=-1 into the function gives f(-1) = (-1)² + 2(-1) + 1 = 1 - 2 + 1 = 0.
  • The y-intercept is 1 (when x=0).
  • The minimum value attained is 0, which is the y-coordinate of the vertex. This minimum occurs at the vertex where the slope is 0.
• f(x) = x² + 8x + 9:
  • For this function, the y-intercept is 9 (when x=0).
  • The axis of symmetry is x = -b/(2a) = -8/(2*1) = -4.
  • The vertex is at x = -4. The y-coordinate of the vertex is f(-4) = (-4)² + 8(-4) + 9 = 16 - 32 + 9 = -7. This value (-7) represents the minimum since a > 0. (Calculation outside of sources, but based on source concepts).
• f(x) = -x² + 1:
  • In this function, a = -1, b = 0, and c = 1.
  • The y-intercept is 1.
  • The axis of symmetry is x = -b/(2a) = -0/(2*(-1)) = 0, which is the y-axis.
  • The vertex is at x = 0. The y-coordinate is f(0) = -(0)² + 1 = 1. The vertex is (0,1).
  • Since a is negative (a < 0), the curve opens downwards. The y-coordinate of the vertex (1) represents the maximum value attained by the function.
  • The graph of this function never intersects the x-axis.
• f(x) = 5x² + 3:
  • This is an example of a parabola that has been shifted upwards.
  • Similar to y=x², this function can only take positive values if the +3 term were absent.
• f(x) = x² + 6x + 8:
  • For this function, a=1, b=6, and c=8.
  • The y-intercept is 8.
  • The axis of symmetry is x = -b/(2a) = -6/(2*1) = -3.
  • The roots (or x-intercepts) are -4 and -2. The value of the function at these points is 0.
  • The vertex is at x = -3. The y-coordinate is f(-3) = (-3)² + 6(-3) + 8 = 9 - 18 + 8 = -1. This is the minimum value since a > 0.
  • Since a > 0 and the vertex value (-1) is negative, the curve opens up and crosses the x-axis at two points, resulting in two real roots.
• x² + 1:
  • This is mentioned in the context of solving the quadratic equation x² + 1 = 0.
  • It’s noted that b=0, so the graph is symmetric about the y-axis.
  • The graph of this function never intersects the x-axis.
  • The discriminant (b² - 4ac) for x² + 1 = 0 is 0² - 4(1)(1) = -4, which is less than 0, indicating no real roots.
• 3x² + 10x - 8:
  • This is given as an example of a quadratic equation in standard form (ax² + bx + c = 0). It was derived from the intercept form with roots 2/3 and -4.
• x² - 4x + 4:
  • This is an example used to demonstrate factoring. The product of the last terms is 4 and the sum of the cross products is -4. It factors into (x-2)(x-2).
  • Setting this equal to 0 (x² - 4x + 4 = 0) shows it has one real root, which is repeated (x=2).
• x² - 25:
  • This example is used to show factoring of a difference of squares. It factors into (x+5)(x-5).
  • Setting this equal to 0 (x² - 25 = 0) gives the roots -5 and 5.
• 9x² - 12x + 4:
  • This is an example used to calculate the discriminant. Here, a=9, b=-12, c=4.
  • The discriminant is b² - 4ac = (-12)² - 4(9)(4) = 144 - 144 = 0.
  • Since the discriminant is 0, it has only one real rational root (repeated).
• 2x² + 16x + 33:
  • This is another example used to calculate the discriminant. Here, a=2, b=16, c=33.
  • The discriminant is b² - 4ac = (16)² - 4(2)(33) = 256 - 264 = -8.
  • Since the discriminant is less than 0, it has no real roots.

These examples illustrate various properties of quadratic functions, such as how the coefficient ‘a’ affects the direction the parabola opens, how to find the y-intercept (c), the importance of the axis of symmetry x = -b/(2a) and the vertex for graphing, how the vertex determines the minimum or maximum value, and the relationship between the function’s zeros and the roots of the corresponding quadratic equation. The slope of a quadratic function is variable and given by 2ax + b, reaching zero at the vertex. Quadratic equations can be solved using methods like graphing (finding x-intercepts), factoring, completing the square, or the quadratic formula. The discriminant (b² - 4ac) within the quadratic formula indicates the number and type of real roots.