You and your friend need to graph quadratic functions of the form y =ax^2 and y =ax^2 + c. Your friend asks you to write some hints to help her graph these types of equations.
a. Explain the role of a.
b. Explain the maximum and minimum.
c. What is the vertex?
d. Explain the role of c.
After reading Reiny's and Anonymous' response, all I can say is Excellent! Excellent! Those two paragraphs are load with good information on the parabola.
C is how much the graph is moved up or down
A is how wide the graph opens up. Large A = small opening and vice versa
The vertex is when the graph changes direction
The maximum or minimum is the vertex depending on which way the parabola opens. if it opens up, its a minimum and opens down its a maximum
y =ax^2 is a parabola with its vertex at (0,0)
if a is positive, the parabola opens upwards and (0,0) is the minimum point of the parabola,
if a is negative, the parabola opens downwards and (0,0) is the maximum point of the parabola.
The constant c causes a vertical shift of y= ax^2 , if c is positive, it moves up, if c is negative, it moves down
e.g.
https://www.wolframalpha.com/input/?i=plot+y+%3D+2x%5E2+,+y+%3D+-2x%5E2,+y+%3D+2x%5E2+%2B+4,+y+%3D+2x%5E2+-+1
a. The role of "a" in the quadratic function y = ax^2 is to determine the shape and direction of the graph. The value of "a" represents the coefficient of x^2 and determines whether the graph opens upwards or downwards. If "a" is positive, the graph will open upwards, and if "a" is negative, the graph will open downwards. The larger the value of "a," the more narrow the graph will be, and the smaller the value of "a," the wider the graph will be.
b. The maximum or minimum point of a quadratic function is known as the vertex. The vertex represents the highest or lowest point on the graph, depending on whether the graph opens upwards (a > 0) or downwards (a < 0). In a quadratic function of the form y = ax^2, the vertex is at the point (0,0) and is also known as the origin.
c. The vertex of a quadratic function is the point on the graph where the maximum or minimum value occurs. The vertex is given by the coordinates (h, k), where h represents the x-value of the vertex and k represents the y-value of the vertex. To find the vertex, you can use the formula h = -b/(2a), where "a" and "b" are the coefficients of the quadratic function (y = ax^2 + bx + c). Substituting the value of h into the equation will give you the corresponding y-value, k.
d. The role of "c" in a quadratic function of the form y = ax^2 + c is to determine the vertical shift of the graph. The value of "c" represents a constant term added to the function and causes the graph to shift up or down. If "c" is positive, the graph will shift upward, and if "c" is negative, the graph will shift downward. The magnitude of "c" determines how far the graph is shifted vertically. Note that "c" does not affect the shape or direction of the graph, only its position along the y-axis.