Quadratic function

Properties of Graphs of Quadratic Functions

1. The graph of a quadratic function f(x) = is called a parabola.

2. If a > 0, the parabola opens upward; if a < 0, the parabola opens downward.

3. As | a| increases, the parabola becomes narrower; as | a| decreases, the parabola becomes wider.

4. The lowest point of a parabola (when a > 0) or the highest point (when a < 0) is called the vertex.

5. The domain of a quadratic function is , because the graph extends indefinitely to the right and to the left. If (h, k) is the vertex of the parabola, then the range of the function is when a > 0 and when a < 0.

6. The graph of a quadratic function is symmetric with respect to a vertical line containing the vertex. This line is called the axis of symmetry. If (h, k) is the vertex of a parabola, then the equation of the axis of symmetry is x = h.

Vertex and Intercepts

1. To determine the vertex of the graph of a quadratic function, f(x) = , we can either:

a) use the method of completing the square to rewrite the function in the form

f(x) = a(x – h) + k. The vertex is (h, k). , or

b) use the formula x = to find the x-coordinate of the vertex; the

y- coordinate of the vertex can be determined by evaluating .

2. If a > 0, then the y-coordinate of the vertex represents the minimum value of the function; if a < 0, then the y-coordinate of the vertex represents the maximum value of the function.

3. To find the y-intercept of the graph of f(x) = , find f(0); to find the x-intercepts, solve the quadratic equation

= 0.

Let’s consider examples:

 

 

 

 

 

 

 

 

 

 

Example 2

Solution

As soon as we knew that a <0 it is mean that the parabola opens downward so it can be d) or b) case. Y –intercept is c and it is negative, so it can be only case b).

 

 

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