The graph of a quadratic function is called a parabola. There is an easy way to tell whether the graph of a quadratic function opens upward or downward: if the leading coefficient is greater than zero, the parabola opens upward, and if the leading coefficient is less than zero, the parabola opens downward.

The sum of the distances from the foci to the vertex is. (a+c)+(a−c)=2a. If (x,y) is a point on the ellipse, then we can define the following variables: d1=the distance from (−c,0) to (x,y)d2=the distance from (c,0) to (x,y) By the definition of an ellipse, d1+d2 d 1 + d 2 is constant for any point (x,y) on the ellipse …

Let’s look at a few key points about these patterns:

1. If the x is squared, the parabola is vertical (opens up or down). If the y is squared, it is horizontal (opens left or right).
2. If a is positive, the parabola opens up or to the right. If it is negative, it opens down or to the left.
3. The vertex is at (h, k).

If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The y -intercept is the point at which the parabola crosses the y -axis. The x -intercepts are the points at which the parabola crosses the x -axis.

Question 177945: A quadratic function has a vertex(-3,0) and opens down. What is the nature of the roots? Since the “quadratic function has a vertex(-3,0)” this means that the vertex lies directly on the x-axis. So this means that the graph crosses the x-axis twice at the same spot (the vertex).

The y- coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively. To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function.

A maximum turning point is a turning point where the curve is concave upwards, f′′(x)<0 f ′ ′ ( x ) < 0 and f′(x)=0 f ′ ( x ) = 0 at the point. A minimum turning point is a turning point where the curve is concave downwards, f′′(x)>0 f ′ ′ ( x ) > 0 and f′(x)=0 f ′ ( x ) = 0 at the point.

Maximum, In mathematics, a point at which a function’s value is greatest. If the value is greater than or equal to all other function values, it is an absolute maximum. If it is merely greater than any nearby point, it is a relative, or local, maximum.

A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). A polynomial of degree n will have at most n – 1 turning points.

The easiest way to find the turning point is when the quadratic is in turning point form (y = a(x – h)2 + k), where (h, k) is the turning point. To get a quadratic into turning point form you need to complete the square.

The turning point will always be the minimum or the maximum value of your graph. To find the turning point of a quadratic equation we need to remember a couple of things: The parabola ( the curve) is symmetrical. If we know the x value we can work out the y value!

In particular, a cubic graph goes to −∞ in one direction and +∞ in the other. So it must cross the x-axis at least once. Furthermore, all the examples of cubic graphs have precisely zero or two turning points, an even number.