Quadratic graphs Today we will be able to construct graphs of quadratic equations that model real life problems.

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Presentation transcript:

Quadratic graphs Today we will be able to construct graphs of quadratic equations that model real life problems

The Graph of a Quadratic Function All parabolas are symmetric with respect to a line called the axis of symmetry. The point where the axis intersects the parabola is the vertex of the parabola. The Graph of a Quadratic Function Leading coefficient is positive. Leading coefficient is negative.

The leading coefficient of ax2 + bx + c is a. y a > 0 opens upward When the leading coefficient is positive, the parabola opens upward and the vertex is a minimum. f(x) = ax2 + bx + c vertex minimum x y vertex maximum When the leading coefficient is negative, the parabola opens downward and the vertex is a maximum. f(x) = ax2 + bx + c a < 0 opens downward Leading Coefficient

Axis of symmetry Formula for: We can also trace to it on the graphing calculator!

The vertex of the graph of f (x) = ax2 + bx + c (a  0) Vertex of a Parabola The vertex of the graph of f (x) = ax2 + bx + c (a  0) Example: Find the vertex of the graph of f (x) = x2 – 10x + 22. f (x) = x2 – 10x + 22 original equation a = 1, b = –10, c = 22 At the vertex, So, the vertex is (5, -3). Vertex of a Parabola

Find the maximum height of the ball. Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is: The path is a parabola opening downward. The maximum height occurs at the vertex. Find the maximum height of the ball. Example: Basketball

The maximum height of the ball is 15 feet. Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is: The path is a parabola opening downward. The maximum height occurs at the vertex. At the vertex, So, the vertex is (9, 15). The maximum height of the ball is 15 feet. Example: Basketball

Graph the following path of a ball Y=-x2+6x Find the Max height After how Seconds does the ball hit the ground.

Plot the points 10 8 6 4 2 0 2 4 6 8 10 Max height is 9 H E I G T The ball hits the ground After 6 seconds 0 2 4 6 8 10 SECONDS

THINGS TO KEEP IN MIND WHEN GRAPHING CHANGE THE VIEWING WINDOW BY CHANGING THE Y MIN AND Y MAX. REMEMBER THAT HEIGHT IS THE VERTICAL AXIS AND TIME IS THE HORIZONTAL AXIS