Quadratic Functions and Modeling

QUADRATIC FUNCTIONS A quadratic function is one of the form: f (x) = ax2 + bx + c where a, b, and c are real numbers with a ≠ 0. The graph of a quadratic function is called a parabola.

STANDARD FORM OF A QUADRATIC EQUATION The graph of is a parabola that has vertex V(h, k) and a vertical axis of symmetry. The parabola opens upward if a is positive; it opens downward if a is negative.

STANDARD FORM OF QUADRATIC FUNCTIONS Every quadratic function given by f (x)  = ax2 + bx + c can be written in the standard form of a quadratic function: f (x) = a(x − h)2 + k, a ≠ 0 The graph of f is a parabola with vertex (h, k). The parabola opens up if a is positive, and it opens down if a is negative. To find the standard form of a quadratic function, use the technique of completing the square.

PUTTING A QUADRATIC FUNCTION IN STANDARD FORM Group the x2 and x terms together. Factor out the coefficient of x2, if necessary. Divide the new coefficient of x by 2. Square the number from Step 3. Add the result of Step 4 inside the parentheses. Take the number from Step 4 times the coefficient in front of the parentheses and subtract it outside the parentheses. Factor the expression in the parentheses as a perfect square.

VERTEX FORMULA The vertex of the graph of f (x)  = ax2 + bx + c is

SUMMARY OF PROPERTIES OF THE GRAPH OF A QUADRATIC FUNCTION f (x)  = ax2 + bx + c, a ≠ 0 The axis of symmetry is the vertical line passing through the vertex of a parabola. Axis of Symmetry: the line x = −b/(2a) Parabola opens up if is a > 0; the vertex is a minimum point. Parabola opens down if is a < 0; the vertex is a maximum point.

MINIMUM AND MAXIMUM VALUE If a is positive, the vertex V(h, k) is the lowest point on the parabola, and the function f has a minimum value f (h) = k. If a is negative, the vertex V(h, k) is the highest point on the parabola, and the function f has a maximum value f (h) = k.

MAXIMUM OR MINIMUM VALUE OF A QUADRATIC FUNCTION If a is positive, then the vertex (h, k) is the lowest point on the graph of f (x)  = a(x − h)2 + k, and the y-coordinate k of the vertex is the minimum value of the function f. If a is negative, then the vertex (h, k) is the highest point on the graph of f (x)  = a(x − h)2 + k, and the y-coordinate k of the vertex is the maximum value of the function f. In either case, the maximum or minimum value is achieved when x = h.

FINDING THE MINIMUM OR MAXIMUM VALUE WITH THE TI-83/84 Enter the quadratic function in Y1. Graph it and set the appropriate viewing window. Press 2nd, CALC and select either 3:minimum or 4:maximum. Use the left arrow to go to a place on the left of the maximum or minimum. Press ENTER. Use the right arrow to go to a place on the right of the maximum or minimum. Press ENTER. Then go near the maximum or minimum. Press ENTER.

THE QUADRATIC FORMULA The quadratic equation ax2 + bx + c = 0 has solutions This equation is called the quadratic formula.

x-INTERCEPTS OF A QUADRATIC FUNCTION If the discriminant b2 − 4ac > 0, then graph of f (x) = ax2 + bx + c has two distinct x-intercepts so it crosses the x-axis in two places. If the discriminant b2 − 4ac = 0, then graph of f (x) = ax2 + bx + c has one x-intercept so it touches the x-axis in at its vertex. If the discriminant b2 − 4ac < 0, then graph of f (x) = ax2 + bx + c has no x-intercept so it does not cross or touch the x-axis.

SOLVING A QUADRATIC EQUATION WITH THE TI-83/84 Enter the quadratic function in Y1. Graph it and set the appropriate viewing window. Press 2nd, CALC and select either 2:zero. Use the left arrow to go to a place on the left of where the graph crosses the x-axis. Press ENTER. Use the right arrow to go to a place on the right of where the graph crosses the x-axis. Press ENTER. Then go near where the graph crosses the x-axis. Press ENTER.

GRAPHS OF QUADRATIC FUNCTIONS

GRAPHS OF QUADRATIC FUNCTIONS

THE POSITION FUNCTION (MODEL) OF AN OBJECT MOVING VERTICALLY If an object is projected straight upward at time t = 0 from a point y0 feet above ground, with an initial velocity v0 ft/sec, then its height above ground after t seconds is given by y(t) = −16t2 + v0 t + y0 We call y0 the initial position and v0 the initial velocity.

EXAMPLE A ball is thrown straight upward from the edge of a roof that is 60 feet above the ground with an initial velocity of 45 feet per second. Write a quadratic model for the height, y(t), of the ball above the ground. What is the maximum height of the ball above the ground? During what time interval will the ball be more than 75 feet above the ground? How long will the ball be in flight?

QUADRATIC POPULATION MODELS A quadratic population model is given by P(t) = P0 + bt + at2. Here, we denote the independent variable by t (time) instead of x, and the constant c by because substitution of t = 0 yields P(0) = P0. We refer to P0 as the initial population.

EXAMPLE The table below shows the census data for Gwinnett County, Georgia from 1960 through 2000. Find the best-fit quadratic model, the SSE, and the average error. Predict the population for Gwinnett County in 2010. Year Population (thousands) 1960 43.5 1970 72.3 1980 166.9 1990 352.9 2000 588.2 Source: US Census Bureau

FINDING THE BEST-FIT QUADRATIC MODEL ON THE TI-83/84 Enter t into L1 and the population into L2. Press STAT and arrow over to CALC. Select 5:QuadReg. Then enter L1 and L2 (or which ever lists you have your x- and y-values stored in). Press L1,L2. Press ENTER.