Given a quadratic equation y = ax^2 + bx + c,
(i) What is the effect of changing the value of the number c on the parabola? In other word, if two parabolas have the same coefficients a, and b, but different values for c, how will their graphs differ?
My answer: If you have different values for c the y interecept would be different and depending on what number you would use for c depends on which way the y intercept would fall.
If you have a = 1, b = 2, x = 3, c = 4, then you would have y = 1(3)^2 + 2(3) + 4 = 9 + 6 + 4 = 19 (3, 19)
If you have a = 1, b = 2, x = 3, c= 0, then you would have y = 1(3)^2 + 2(3) + 0 = 9 + 6 = 15 (3, 15)
(ii) What is the effect of decreasing the value a toward 0 on the graph of the parabola?
My answer: The parabla sides would stretch out to one side,
In order to explain the effect of changing the value of the number c on the parabola, let's start by looking at the general form of a quadratic equation: y = ax^2 + bx + c.
The number c in this equation represents the constant term, which is the y-intercept of the parabola. It tells us where the parabola intersects the y-axis when x = 0.
If you have two parabolas with the same coefficients a and b, but different values for c, their graphs will differ in terms of their y-intercepts. When you increase or decrease the value of c, you are essentially shifting the position of the parabola up or down on the y-axis.
For example, let's consider two quadratic equations: y = x^2 + 2x + 4 and y = x^2 + 2x + 0.
In the first equation, with c = 4, the y-intercept is (0, 4). This means that the parabola intersects the y-axis at y = 4.
In the second equation, with c = 0, the y-intercept is (0, 0). This means that the parabola intersects the y-axis at y = 0.
So, changing the value of c affects the y-intercept of the parabola. Higher values of c will shift the parabola upward, while lower values of c will shift the parabola downward.
Moving on to the effect of decreasing the value of a toward 0 on the graph of the parabola, let's consider the general form of a quadratic equation again: y = ax^2 + bx + c.
The coefficient a is responsible for determining the shape of the parabola. When a is positive, the parabola opens upward, forming a U-shape. When a is negative, the parabola opens downward, forming an inverted U-shape.
When you decrease the value of a toward 0, the parabola becomes more and more stretched out horizontally. It starts to resemble a straight line, as the coefficient of x^2 becomes insignificant.
For example, let's consider two quadratic equations: y = x^2 + 2x + 4 and y = 0.1x^2 + 2x + 4.
In the first equation, with a = 1, the parabola has a relatively normal U-shape.
In the second equation, with a = 0.1, the parabola is significantly flatter and wider. It is stretched out horizontally and resembles more of a line.
In summary, decreasing the value of a toward 0 stretches out the parabola's sides horizontally, making it flatter and wider.