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,
To discuss the effect of changing the value of c on the parabola y = ax^2 + bx + c, let's first understand the general properties of a quadratic equation.
A quadratic equation represents a parabola on a graph, which is a U-shaped curve. The coefficient a determines the direction and width of the parabola, and the coefficients b and c determine the position of the vertex of the parabola. The vertex represents the lowest or highest point on the parabola, depending on the value of a.
Now, let's specifically consider the effect of changing the value of c in two parabolas with the same coefficients a and b.
If we have y = ax^2 + bx + c1 and y = ax^2 + bx + c2, where c1 and c2 are different values, the parabolas will have different y-intercepts. The y-intercept is the point at which the parabola crosses the y-axis, where x = 0.
By substituting x = 0 into the equations, we can determine their y-intercepts. For the first equation, y = a(0)^2 + b(0) + c1 = c1. So, the y-intercept is (0, c1). Similarly, for the second equation, the y-intercept is (0, c2).
Therefore, changing the value of c shifts the parabola up or down on the y-axis, resulting in different y-intercepts. The higher the value of c, the higher the parabola will intersect the y-axis, and vice versa.
To visualize this, consider two examples:
1. If a = 1, b = 2, x = 3, and c1 = 4, the equation y = 1(3)^2 + 2(3) + 4 simplifies to y = 9 + 6 + 4 = 19. So, the parabola intersects the y-axis at (3, 19).
2. If a = 1, b = 2, x = 3, and c2 = 0, the equation y = 1(3)^2 + 2(3) + 0 simplifies to y = 9 + 6 = 15. So, the parabola intersects the y-axis at (3, 15).
As for the effect of decreasing the value of a towards 0 on the graph of the parabola, when a is small or becomes 0, the parabola will become flatter and wider. It will resemble more of a straight line than a U-shaped curve. This means that as a approaches 0, the parabola loses its curvature and becomes closer to a straight line.