Quadratic Functions

3.1

Graphs of quadratic functions

Standard (vertex) formGeneral form

Standard (vertex) form f(x)=a(x-h)2 +k

◦Open up or down? Max or min?◦Vertex is (h, k)◦Find x-intercepts◦Find y-intercept by figuring f(0)◦Plot the points, add more points if necessary (remember symmetry)

◦The domain is (-∞,∞)

Graphs of quadratic functions

U/D? Vertex X-int. Y-int Graph it Domain Range

Standard form: 2( ) 2 3 8f x x

U/D? Vertex X-int. Y-int Graph it Domain Range

Standard form: 2( ) 1 4f x x

U/D? Vertex X-int. Y-int Graph it Domain Range

Standard form: 2( ) 3 1f x x

U/D? Vertex X-int. Y-int Graph it Domain Range

Standard form: 2( ) 2 1f x x

General form f(x)=ax2 +bx + c◦Open up or down? Max or min?◦x- value of vertex is –b/2a. Plug in for y-value.

◦Find x-intercepts◦Find y-intercept by figuring f(0)◦Plot the points, add more points if necessary (remember symmetry)

◦The domain is (-∞,∞)

Graphs of quadratic functions

U/D? Vertex X-int. Y-int Graph it Domain Range

General form: 2( ) 2 1f x x x

U/D? Vertex X-int. Y-int Graph it Domain Range

General form: 2( ) 4 1f x x x

Applications of quadratic functions

Maximizing and minimizing

The function f(x) = 0.4x2 – 36x + 1000 models the number of accidents, f(x), per 50 million miles driven, for drivers x years old, where 16 ≤ x ≤ 74. What is the age of a driver having the least number of car accidents? What is the minimum number of car accidents per 50 million miles driven?

Applications of quadratic functions

Maximizing or minimizing quadratic functions◦What is the quantity to be maximized/minimized?

◦Express the quantity as a function in one variable.

◦Write the function in general form.◦Find the vertex of the function.◦Use the vertex to help answer the question.

Applications of quadratic functions

Among all pairs of numbers whose difference is 10, find a pair whose product is as small as possible. What is the minimum product?

◦Minimize what?◦Write with one variable:◦Put in general form:◦Find vertex:◦Answer question:

Applications of quadratic functions

Among all pairs of numbers whose difference is 8, find a pair whose product is as small as possible. What is the minimum product?

◦Minimize what?◦Write with one variable:◦Put in general form:◦Find vertex:◦Answer question:

Applications of quadratic functions

You have 100 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

◦Maximize what?◦Write with one variable:◦Put in general form:◦Find vertex:◦Answer question:

Applications of quadratic functions

You have 120 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

◦Maximize what?◦Write with one variable:◦Put in general form:◦Find vertex:◦Answer question:

Applications of quadratic functions