Graphing Quadratic Functions Adapted from Walch Education
Slide 3
Key Concepts The graph of a quadratic is U-shaped and called a
parabola. The extremum of a graph is the function value that
achieves either a maximum or a minimum. The maximum is the largest
y-value of a quadratic, and the minimum is the smallest y-value of
a quadratic. 5.6.1: Graphing Quadratic Functions2
Slide 4
Key Concepts, continued If a > 0, the parabola is concave up
and the quadratic has a minimum. If a < 0, the parabola is
concave down and the quadratic has a maximum. The extreme values of
a quadratic occur at the vertex, the point at which the curve
changes direction. If the quadratic equation is given in standard
form, the vertex can be found by identifying the x- coordinate of
the vertex using 5.6.1: Graphing Quadratic Functions3
Slide 5
Key Concepts, continued Substitute the value of the
x-coordinate into the quadratic equation to find the y-coordinate
of the vertex. The vertex of a quadratic function is The vertex
form of a quadratic function is f(x) = a(x h) 2 + k, where the
coordinate pair (h, k) is the location of the vertex. 5.6.1:
Graphing Quadratic Functions4
Slide 6
Intercepts The intercept of a graph is the point at which a
line intercepts the x- or y-axis. The y-intercept of a function is
the point at which the graph crosses the y-axis. This occurs when x
= 0. The y-intercept is written as (0, y). The x-intercepts of a
function are the points at which the graph crosses the x-axis. This
occurs when y = 0. The x-intercept is written as (x, 0). 5.6.1:
Graphing Quadratic Functions5
Slide 7
Zeros The zeros of a function are the x-values for which the
function value is 0. The intercept form of the quadratic function,
written as f(x) = a(x p)(x q), where p and q are the zeros of the
function, can be used to identify the x- intercepts. Set the
factored form equal to 0. Then set each factor equal to 0. As long
as the coefficients of x are 1, the x-intercepts are located at (r,
0) and (s, 0). These values for x are the roots, or the solutions,
of the quadratic equation. 5.6.1: Graphing Quadratic
Functions6
Slide 8
Symmetry Parabolas are symmetrical ; that is, they have two
identical parts when rotated around a point or reflected over a
line. This line is the axis of symmetry, the line through the
vertex of a parabola about which the parabola is symmetric. The
equation of the axis of symmetry is Symmetry can be used to find
the vertex of a parabola if the vertex is not known. 5.6.1:
Graphing Quadratic Functions7
Slide 9
X-coordinate of the vertex If you know the x-intercepts of the
graph or any two points on the graph with the same y-value, the x-
coordinate of the vertex is the point halfway between the values of
the x-coordinates. For x-intercepts (r, 0) and (s, 0), the
x-coordinate of the vertex is 5.6.1: Graphing Quadratic
Functions8
Practice # 1 Given the function f(x) = 2x 2 + 16x 30, identify
the key features of the graph: the extremum, vertex, x-
intercept(s), and y-intercept. Then sketch the graph. 5.6.1:
Graphing Quadratic Functions10
Slide 12
Determine the extremum of the graph The extreme value is a
minimum when a > 0. It is a maximum when a < 0. Because a =
2, the graph opens downward and the quadratic has a maximum. 5.6.1:
Graphing Quadratic Functions11
Slide 13
Determine the vertex of the graph The maximum value occurs at
the vertex. The vertex is of the form Use the original equation
f(x) = 2x 2 + 16x 30 to find the values of a and b in order to find
the x- value of the vertex. 5.6.1: Graphing Quadratic
Functions12
Slide 14
continued The x-coordinate of the vertex is 4. 5.6.1: Graphing
Quadratic Functions13 Formula to find the x-coordinate of the
vertex of a quadratic Substitute 2 for a and 16 for b. x =
4Simplify.
Slide 15
continued Substitute 4 into the original equation to find the
y- coordinate. The y-coordinate of the vertex is 2. The vertex is
located at (4, 2). 5.6.1: Graphing Quadratic Functions14 f(x) = 2x
2 + 16x 30Original equation f(4) = 2(4) 2 + 16(4) 30Substitute 4
for x. f(4) = 2 Simplify.
Slide 16
Determine the x- intercept(s) of the graph Since the vertex is
above the x-axis and the graph opens downward, there will be two
x-intercepts. Factor the quadratic and set each factor equal to 0.
5.6.1: Graphing Quadratic Functions15
Slide 17
continued The x-intercepts are (3, 0) and (5, 0). 5.6.1:
Graphing Quadratic Functions16 f(x) = 2x 2 + 16x 30Original
equation f(x) = 2(x 2 8x + 15) Factor out the greatest common
factor. f(x) = 2(x 3)(x 5)Factor the trinomial. 0 = 2(x 3)(x 5) Set
the factored form equal to 0 to find the intercepts. x 3 = 0 or x 5
= 0 Set each factor equal to 0 and solve for x. x = 3 or x =
5Simplify.
Slide 18
Determine the y-intercept of the graph The y-intercept occurs
when x = 0. Substitute 0 for x in the original equation. The
y-intercept is (0, 30). When the quadratic equation is written in
standard form, the y-intercept is c. 5.6.1: Graphing Quadratic
Functions17 f(x) = 2x 2 + 16x 30Original equation f(0) = 2(0) 2 +
16(0) 30Substitute 0 for x. f(0) = 30Simplify.
Slide 19
Graph the function Use symmetry to identify additional points
on the graph. The axis of symmetry goes through the vertex, so the
axis of symmetry is x = 4. For each point to the left of the axis
of symmetry, there is another point the same distance on the right
side of the axis and vice versa. The point (0, 30) is on the graph,
and 0 is 4 units to the left of the axis of symmetry. 5.6.1:
Graphing Quadratic Functions18
Slide 20
continued The point that is 4 units to the right of the axis is
8, so the point (8, 30) is also on the graph. Determine two
additional points on the graph. Choose an x-value to the left or
right of the vertex and find the corresponding y-value. 5.6.1:
Graphing Quadratic Functions19 f(x) = 2x 2 + 16x 30Original
equation f(1) = 2(1) 2 + 16(1) 30Substitute 1 for x. f(1) =
16Simplify.
Slide 21
continued An additional point is (1, 16). (1, 16) is 3 units to
the left of the axis of symmetry. The point that is 3 units to the
right of the axis is 7, so the point (7, 16) is also on the graph.
Plot the points and join them with a smooth curve. 5.6.1: Graphing
Quadratic Functions20
Slide 22
Graph the function 5.6.1: Graphing Quadratic Functions21
Slide 23
Your turn: Given the function f(x) = 2(x + 1)(x + 5), identify
the key features of its graph: the extremum, vertex, x-
intercept(s), and y-intercept. Then sketch the graph. 5.6.1:
Graphing Quadratic Functions22
