Download CHAPTER 8

Unpacking the Standards
Understanding the standards and the vocabulary terms in the standards will
help you know exactly what you are expected to learn in this chapter.
What It Means For You Lessons 8-1, 8-2, 8-3, 8-4, 8-5, 8-9, 8-10
Graph … quadratic functions and show
intercepts, maxima, and minima.
The graph of a quadratic function has key features that are helpful
when interpreting a real-world quadratic model: the intercepts and the
maximum or minimum value.
Key Vocabulary
EXAMPLE Graph of y = x
​ 2​ ​ + 2x - 3
quadratic function (función
cuadrática) A function that can be
written in the form f​(x)​= a​x2​ ​+
bx + c, where a, b, and c are real
numbers and a ≠ 0, or in the form
2
f​(x)​= a​​(x - h)​​ ​+ k, where a, h,
and k are real numbers and a ≠ 0.
x-intercept (intersección con el eje
x) The x-coordinate(s) of the point(s)
where a graph intersects the x-axis.
y-intercept (intersección con el eje
y) The y-coordinate(s) of the point(s)
where a graph intersects the y-axis.
4
The x-intercepts
are -3 and 1.
y
x
0
(-3, 0)
The minimum
value is -4.
(1, 0)
4
(0, -3)
(-1, -4)
The y-intercept is -3.
maximum/minimum value of a
function (máximo/mínimo de una
función) The y-value of the highest/
lowest point on the graph of the
function.
CC.9-12.F.BF.3
Identify the effect on the graph of
replacing f(x) by f(x) + k, k f(x), f(kx),
and f(x + k) for specific values of k
(both positive and negative); …
What It Means For You
Lessons 8-1, 8-4
You can change a function by adding or multiplying by a constant. The
result will be a new function that is a transformation of the original
function.
EXAMPLE Compression and Stretch/Reflection of f(x)
y
4
2
f(x) = x
Key Vocabulary
function notation (notación de
función) If x is the independent
variable and y is the dependent
variable, then the function notation
for y is f(x), read “f of x,” where f
names the function.
2
g(x) = 1 x2
2
-4
-2
x
0
2
4
-2
h(x) = -3x2
-4
Chapter 8
408
Quadratic Functions and Equations
© Houghton Mifflin Harcourt Publishing Company
chapter 8
CC.9-12.F.IF.7a
CC.9-12.A.REI.4b
Solve quadratic equations by inspection
(e.g., for ​x2​ ​= 49), taking square roots,
completing the square, the quadratic
formula and factoring, as appropriate to
the initial form of the equation. ...
Key Vocabulary
quadratic equation (ecuación
cuadrática) An equation that can be
written in the form a​x2​ ​+ bx + c = 0,
where a, b, and c are real numbers and
a ≠ 0.
completing the square (completar el
cuadrado) A process used to form a
perfect-square trinomial. To complete
2
t he square of ​x​2​+ bx, add ​​ _
​ b ​  ​​ ​.
2
Quadratic Formula (fórmula cuadrática)
Lessons 8-6, 8-7, 8-8, 8-9, 8-10
Knowing how to solve quadratic equations gives you tools to
understand many situations, including the laws of motion. Recognizing
the best solution method for a situation allows you to work efficiently.
EXAMPLE Solving a Quadratic Equation
The height h in feet of a baseball leaving a certain batter’s bat is
h​(t)​= -16​t2​ ​+ 63t + 4, where t is in seconds. When does the ball
hit the ground?
-16​t​2​+ 63t + 4 = 0
The ball hits the ground when h = 0.
-1​(16t + 1)​​(t - 4)​= 0 You can factor the equation.
1  ​or t = 4
t = -​ __
16
The factors give these solutions.
The ball hits the ground in 4 seconds. (The negative value is not
reasonable in the real-world context.)
The formula
 
  2​ ​- 4ac ​ 
-b ± ​ √  
​b
x = ​ __
      
​
2a
which gives solutions, or roots, of
equations in the form ​ax​2​+ bx + c = 0,
where a, b, and c are real numbers
and a ≠ 0.
chapter 8
© Houghton Mifflin Harcourt Publishing Company; Photo credit: © Rim Light/PhotoLink/Photodisc/GettyImages
()
What It Means For You
What It Means For You
CC.9-12.A.REI.7
Solve a simple system consisting of a
linear equation and a quadratic equation
in two variables algebraically and
graphically.
Key Vocabulary
linear equation in two variables (ecuación lineal en dos variables) An
equation that can be written in the
form Ax + By = C where A, B, and
C are constants and A and B are not
both 0.
Solving a system of equations in two variables involves finding the
ordered pair or pairs of values that make both equations true. You can
do this algebraically or graphically.
EXAMPLE Solving a System of Equations
 ⎧ y = ​x​2​- 2x - 3
​ ​ 
⎨ 
⎩  y = -x - 1
​x​2​- 2x - 3 = -x - 1
​x​2​- x - 2 = 0
(x - 2)(x + 1) = 0
Chapter 8
Lessons 8-10
Solutions: (2, -3), (-1, 0)
409
2
y
x
-2
0
2
4
-2
-4
Quadratic Functions and Equations
Key Vocabulary
completing the square (completar el cuadrado) A process used to form a perfect-square trinomial.
(  )
2
b
To complete the square of ​x2​ ​+ bx, add ​​ __
​ 2 ​  ​​ ​.
discriminant (discriminante) The discriminant of the quadratic equation a​x2​ ​+ bx + c = 0
2
is b
​  ​​– 4ac.
maximum/minimum value of a function (máximo/mínimo de una función) The y-value of the
highest/lowest point on the graph of the function.
parabola (parábola) The shape of the graph of a quadratic function.
2
quadratic equation (ecuación cuadrática) An equation that can be written in the form a​x ​​+ bx +
c = 0, where a, b, and c are real numbers and a ≠ 0.
2
√
​b​ ​- 4ac ​ 
     ​ 
which gives solutions, or
Quadratic Formula (fórmula cuadrática) The formula x = _____________
​ -b ± ​ 2a
roots, of equations in the form a​x2​ ​+ bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
chapter 8
quadratic function (función cuadrática) A function that can be written in the form f(x) = a​x2​ ​+
bx + c, where a, b, and c are real numbers and a ≠ 0, or in the form f(x) = a(x – h​)​2​+ k, where a, h,
and k are real numbers and a ≠ 0.
vertex of a parabola (vértice de una parábola) The highest or lowest point on the parabola.
x-intercept (intersección con el eje x) The x-coordinate(s) of the point(s) where a graph intersects the
x-axis.
y-intercept (intersección con el eje y) The y-coordinate(s) of the point(s) where a graph intersects the
y-axis.
Zero Product Property (Propiedad del producto cero) For real numbers p and q, if pq = 0, then
p = 0 or q = 0.
The Common Core Standards for Mathematical Practice describe varieties
of expertise that mathematics educators at all levels should seek to develop
in their students. Opportunities to develop these practices are integrated
throughout this program.
1. M
ake sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. C
onstruct viable arguments and critique the reasoning of others.
4. M
odel with mathematics.
5. Use appropriate tools strategically.
6. A
ttend to precision.
7. L ook for and make use of structure.
8. L ook for and express regularity in repeated reasoning
Chapter 8
410
Quadratic Functions and Equations