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Mathematics A
1. Pure Mathematics
1.2 Algebra and Functions
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Cards (213)
What are the two types of solutions for algebraic equations and inequalities?
Specific values and ranges
To solve equations like
f
(
x
)
=
f(x) =
f
(
x
)
=
g
(
x
)
g(x)
g
(
x
)
, you need to find the value(s) of
x
x
x
for which both sides are equal
Inequalities
define a range of values that satisfy the condition.
What does the domain of an algebraic function
f
(
x
)
f(x)
f
(
x
)
represent?
All possible input values
The domain of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
\sqrt{x}
x
includes all real numbers.
False
Match the graph feature with its description:
Intercepts ↔️ Points where the graph crosses the axes
Turning Points ↔️ Maximum or minimum points
Asymptotes ↔️ Lines the graph approaches but never touches
Steps to find the x-intercepts of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
+
x^{2} - 4x +
x
2
−
4
x
+
3
3
3
1️⃣ Set
y
=
y =
y
=
0
0
0
2️⃣ Solve
x
2
−
4
x
+
x^{2} - 4x +
x
2
−
4
x
+
3
=
3 =
3
=
0
0
0
3️⃣ Factor the equation
(
x
−
1
)
(
x
−
3
)
=
(x - 1)(x - 3) =
(
x
−
1
)
(
x
−
3
)
=
0
0
0
4️⃣ Find the values
x
=
x =
x
=
1
1
1
and
x
=
x =
x
=
3
3
3
What are the key features to identify when sketching algebraic functions?
Intercepts and asymptotes
Intercepts are where the graph crosses the x-axis and
y-axis
Asymptotes
are lines that the graph approaches but never touches.
Match the feature with its description for
f
(
x
)
=
f(x) =
f
(
x
)
=
1
x
\frac{1}{x}
x
1
:
X-intercept ↔️ None
Y-intercept ↔️ (0, undefined)
Asymptotes ↔️ x=0, y=0
What are the asymptotes of the graph f(x) = \frac{1}{x}</latex>?
x=0 and y=0
The graph of
f
(
x
)
=
f(x) =
f
(
x
)
=
1
x
\frac{1}{x}
x
1
has no intercepts.
In which quadrants does the graph of
f
(
x
)
=
f(x) =
f
(
x
)
=
1
x
\frac{1}{x}
x
1
lie?
Quadrants I and III
Understanding graph features helps in analyzing its
behavior
Transformations such as shifts, stretches, compressions, or
reflections
can affect the graph of a function.
What does the transformation
f
(
x
−
a
)
f(x - a)
f
(
x
−
a
)
represent in terms of the graph?
Horizontal shift
A vertical shift of
f
(
x
)
f(x)
f
(
x
)
by
b
b
b
units is represented by f(x) + b</latex>
Asymptotes are lines that the graph approaches but never
touches
Intercepts are points where the graph crosses the
axes
.
Match the transformation with its equation:
Horizontal Shift ↔️
f
(
x
−
a
)
f(x - a)
f
(
x
−
a
)
Vertical Shift ↔️
f
(
x
)
+
f(x) +
f
(
x
)
+
b
b
b
Horizontal Stretch ↔️
f
(
x
a
)
f(\frac{x}{a})
f
(
a
x
)
Vertical Stretch ↔️
a
⋅
f
(
x
)
a \cdot f(x)
a
⋅
f
(
x
)
Order the steps to transform
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
to
f
(
x
−
3
)
=
f(x - 3) =
f
(
x
−
3
)
=
(
x
−
3
)
2
(x - 3)^{2}
(
x
−
3
)
2
1️⃣ Identify the type of transformation
2️⃣ Substitute
x
x
x
with
x
−
3
x - 3
x
−
3
3️⃣ Apply the substitution in the equation
4️⃣ Simplify the equation if necessary
What is the effect of
f
(
x
−
c
)
f(x - c)
f
(
x
−
c
)
on the graph of
f
(
x
)
f(x)
f
(
x
)
?
Horizontal shift
The transformation
f
(
x
+
c
)
f(x + c)
f
(
x
+
c
)
shifts the graph to the left if
c
>
0
c > 0
c
>
0
.
The transformation
a
f
(
x
)
af(x)
a
f
(
x
)
scales the graph vertically by a factor of a
What is the effect of
−
f
(
x
)
- f(x)
−
f
(
x
)
on the graph of
f
(
x
)
f(x)
f
(
x
)
?
Vertical reflection
What is the effect of multiplying the function by -1, i.e.,
−
f
(
x
)
- f(x)
−
f
(
x
)
?
Reflection across x-axis
What is the effect of adding a constant to the input, e.g.,
f
(
x
+
c
)
f(x + c)
f
(
x
+
c
)
?
Horizontal shift left by c
Vertical scaling involves multiplying the function by a constant, e.g.,
a
f
(
x
)
af(x)
a
f
(
x
)
.
Multiplying the input by a constant, e.g.,
f
(
a
x
)
f(ax)
f
(
a
x
)
, results in horizontal scaling.
What is the effect of adding a constant to the function, e.g.,
f
(
x
)
+
f(x) +
f
(
x
)
+
c
c
c
?
Vertical shift up by c
What is the effect of multiplying the input by a constant, e.g.,
f
(
a
x
)
f(ax)
f
(
a
x
)
?
Horizontal scaling
Multiplying the input by -1, e.g.,
f
(
−
x
)
f( - x)
f
(
−
x
)
, results in reflection across the y-axis.
Order the key transformations of algebraic functions from simple to complex effects on the graph.
1️⃣ Vertical shifts
2️⃣ Horizontal shifts
3️⃣ Vertical scaling
4️⃣ Horizontal scaling
5️⃣ Reflection
What is the effect of multiplying the function by a constant, e.g.,
a
f
(
x
)
af(x)
a
f
(
x
)
?
Vertical scaling
What is the effect of adding a constant to the function, e.g.,
f
(
x
)
+
f(x) +
f
(
x
)
+
c
c
c
?
Vertical shift up by c
Multiplying the input by a constant, e.g.,
f
(
a
x
)
f(ax)
f
(
a
x
)
, results in horizontal scaling.
Multiplying the input by -1, e.g.,
f
(
−
x
)
f( - x)
f
(
−
x
)
, results in reflection across the y-axis.
Match the transformation with its effect on the graph:
f
(
x
)
+
f(x) +
f
(
x
)
+
c
c
c
↔️ Vertical shift up by c
f
(
x
−
c
)
f(x - c)
f
(
x
−
c
)
↔️ Horizontal shift right by c
−
f
(
x
)
- f(x)
−
f
(
x
)
↔️ Reflection across x-axis
f
(
a
x
)
f(ax)
f
(
a
x
)
↔️ Horizontal scaling
What is the effect of adding a constant to the function, e.g., f(x) + c</latex>?
Vertical shift up by c
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