4.1A Graph Quadratic Functions in Standard Form
Goal · Graph quadratic functions.
VOCABULARY:
(1) Quadratic function: A function that can be written in the standard form y = ax2 + bx+ c where a ¹ 0
(2) Parabola: The U-shaped graph of a quadratic function
(3) Vertex: The lowest or highest point on a parabola
(4) Axis of symmetry: The vertical line that divides the parabola in half and passes through the vertex.
(5) Minimum and maximum value: For y= ax2 + bx+ c, the vertex's y-coordinate is the minimum value of the function if a 0 and its maximum value if a 0.
PARENT FUNCTION FOR QUADRATIC FUNCTIONS
The parent function for the family of all quadratic functions is f(x) = __x2__ . The graph is shown below
For f(x) = ax2, and for any quadratic function g(x)= ax2 + bx + c where b = 0, the vertex lies on the __y-axis__ and the axis of symmetry is x = _0_.
Example 1: Graph a function of the form y = ax2 + c
Graph y = -2x2 + 2. Compare the graph with the graph of y = x2.
1. Make a table of values for y = -2x2 + 2.
x / -2 / -1 / 0 / 1 / 2y / _-6_ / _0_ / _2_ / _0_ / _-6_
2. Plot the points from the table.
3. Draw a smooth __curve__ through the points.
4. Compare the graphs of y = -2x2 + 2 and y = x2. Both graphs have the same __axis of symmetry__. However, the graph of y = -2x2 + 2 opens _down_ and is _narrower_ than the graph of y = x2. Also, its vertex is _2_units higher.
PROPERTIES OF THE GRAPH OF y = ax2 + bx + c
Characteristics of the graph of y = ax2 + bx + c:
· The graph opens up if a __ 0 and opens down if a __ 0.
· The graph is narrower than the graph of y = x2 if | a | __ 1 and wider if | a | __ 1.
· The axis of symmetry is x = and the vertex has x-coordinate .
· The y-intercept is _c_. So, the point (0, _c_) is on the parabola.
Example 2: Graph a function of the form y = ax2 + bx + c
Graph y = -x2 + 4x - 3.
1. Identify the coefficients of the function. The coefficients are a = _-1_, b = _4_, and
c = _-3_. Because a __ 0, the parabola opens _down_.
2. Find the vertex. First, calculate the x-coordinate.
x = = = _2_
Then find the y-coordinate.
y = __-(2)2 + 4(2) - 3__ = _1_
The vertex is (_2_, _1_). Plot this point.
3. Draw the axis of symmetry x = _2_.
4. Identify the y-intercept c, which is _-3_.
Plot the point (0, _-3_). Then reflect this point in the axis of symmetry to plot another point (4, _-3_).
5. Evaluate the function for another value of x, such as x = 1.
y = __-(1)2 + 4(1) - 3__ = _0_
Plot the point (1, _0_) and its reflection (3, _0_) in the axis of symmetry.
6. Draw a parabola through the plotted points.
MINIMUM AND MAXIMUM VALUES
Words For y = ax2 + bx + c, the vertex's y-coordinate is the minimum value of the function if a __ 0 and the maximum value if a __ 0.
Example 3: Find the minimum or maximum value
Tell whether the function y = -3x2 + 12x - 6 has a minimum value or a maximum value. Then find the minimum or maximum value.
Solution
Because a __ 0, the function has a __maximum__ value. To find it, calculate the coordinates of the vertex.
x = = = _2_
y = __-3(2)2 + 12(2) - 6__ = _6_
The maximum value is y = _6_ .
You Try: Complete the following exercises.
1. y = 1/3 x2 + 3
2. Graph the function. Label the vertex and axis of symmetry. y = x2 - 4x + 2