## Part 1: Understanding the Concept of Functions

Welcome to MKSprep, your premier SAT preparation center in Kathmandu, Nepal. In this course, we’ll be diving deep into the concept of Functions and Graphs, essential elements of the SAT Math section.

### What is a Function?

In mathematics, a function is a rule that relates inputs to outputs in such a way that each input corresponds to exactly one output. The set of all possible inputs is called the domain, and the set of corresponding outputs is the range.

A function is typically represented as f(x), where ‘x’ is the input, and ‘f(x)’ represents the output. The most common types of functions you’ll encounter on the SAT are:

• Linear functions (f(x) = mx + b).
• Quadratic functions (f(x) = ax² + bx + c).
• Exponential functions (f(x) = a * b^x).

### Visualizing Functions with Graphs

One of the most effective ways to understand and interpret functions is through graphical representation. A function graph shows the relationship between the input (on the x-axis) and the output (on the y-axis). This graphical perspective can help us see patterns, identify key function attributes, and solve problems more intuitively.

For example, the linear function graph is a straight line, while the graph of a quadratic function forms a parabola.

The Graph of Function

### Practice

Try sketching the graph for these functions:

1. f(x) = 2x + 3
2. f(x) = x² – 4x + 2

In the next part, we’ll delve into the important aspects of linear functions, their properties, and how to graph them.

[Next: Linear Functions and Their Graphs]

## Part 2: Linear Functions and Their Graphs

Welcome back to MKSprep’s SAT preparation course in Kathmandu, Nepal. This section explores Linear Functions, their properties, and how to graph them effectively.

### Understanding Linear Functions

A linear function has the general form y = mx + b, where ‘m’ is the slope of the line, and ‘b’ is the y-intercept. This function forms a straight line when graphed.

The slope ‘m’ tells us the steepness of the line and the direction it travels in (upwards for positive slopes and downwards for negative slopes). The y-intercept ‘b’ is the point at which the line crosses the y-axis.

### Graphing Linear Functions

When graphing a linear function, start by plotting the y-intercept on the graph. Then use the slope to find additional points on the line. Remember, the slope is often given as a fraction (rise/run), where ‘rise’ is the change in y (vertical), and ‘run’ is the change in x (horizontal).

figure

Graph of a Linear Function

### Practice

Consider the function f(x) = 3x – 2:

1. The y-intercept is at (0, -2). Plot this point on your graph first.
2. The slope is 3, which we can consider as 3/1 for graphing purposes. From the y-intercept, go up 3 units (rise) and to the right 1 unit (run).
3. Mark this new point and draw a straight line connecting your points. Extend this line in both directions.

Congratulations! You’ve graphed a linear function. In the next part of our course, we’ll explore quadratic functions and their unique properties and graphical characteristics.

[Next: Quadratic Functions and Their Graphs]

## Part 3: Quadratic Functions and Their Graphs

Continuing with MKSprep’s SAT preparation course, we now move on to quadratic functions, a central topic in SAT Math. In this section, we’ll look at the properties of quadratic functions and how to graph them.

A quadratic function has the general form f(x) = ax² + bx + c, where ‘a’, ‘b’, and ‘c’ are constants, and ‘a’ is not equal to zero. The graph of a quadratic function is a parabola. If ‘a’ is positive, the parabola opens upwards; if ‘a’ is negative, it opens downwards.

The vertex of the parabola is the highest or lowest point (depending on whether the parabola opens up or down) and the line of symmetry is a vertical line passing through the vertex.

Here’s a basic process to graph a quadratic function:

1. Identify the vertex. If the quadratic is in the form f(x) = a(x-h)² + k, the vertex is at (h, k).
2. Plot the vertex on your graph.
3. Identify the y-intercept (the value of ‘c’) and plot this point.
4. Use the line of symmetry to find other points on the parabola.
5. Draw the parabola connecting the points.

figure

### Practice

Consider the function f(x) = (x – 3)² + 1:

1. The vertex is at (3, 1). Plot this point on your graph.
2. This parabola opens upwards (since the coefficient of x² is positive).
3. Find the y-intercept by setting x = 0: f(0) = (0 – 3)² + 1 = 10.
4. Use the line of symmetry to find other points on the parabola and draw the curve.

Now, you’ve graphed a quadratic function! Stay tuned for our next session where we’ll introduce exponential functions and their graphing techniques.

[Next: Exponential Functions and Their Graphs]

## Part 3: Quadratic Functions and Their Graphs

Continuing our SAT preparation course here at MKSprep, Kathmandu, Nepal, we will now turn our focus towards quadratic functions. In this section, we’ll learn a quadratic function, its properties, and how to graph it accurately.

A quadratic function is a function that can be described by an equation of the form f(x) = ax² + bx + c, where ‘a’, ‘b’, and ‘c’ are constants and ‘a’ is not zero. The highest power of the variable (x) is 2, hence the name quadratic.

A quadratic function represents a parabola when graphed. The parabola’s direction depends on the coefficient ‘a’ sign. If ‘a’ is positive, the parabola opens upwards. If ‘a’ is negative, it opens downwards.

1. Identify the vertex. The vertex is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards.
2. Plot the vertex on your graph.
3. Identify and plot the y-intercept, which is the value of the function when x = 0 (this is equal to ‘c’).
4. Use symmetry about the vertex to plot additional points on the graph.
5. Connect the points to draw the parabola.

figure

### Practice

Consider the function f(x) = -2(x + 1)² + 3:

1. The vertex of this function is at (-1, 3). Plot this point on your graph.
2. The function opens downwards because ‘a’ is negative.
3. The y-intercept is f(0) = -2(0 + 1)² + 3 = 1. Plot this point on your graph.
4. Use symmetry and plot additional points on the graph.
5. Connect the points to create your graph.

In the next part, we’ll look at how to solve and graph exponential functions, which are integral to the SAT math syllabus.

[Next: Exponential Functions and Their Graphs]

## Part 4: Exponential Functions and Their Graphs

As we continue our SAT preparation course at MKSprep in Kathmandu, Nepal, we now shift our focus to exponential functions. These functions play a vital role in many mathematical concepts and real-world applications.

### Understanding Exponential Functions

An exponential function is a mathematical function of the form f(x) = b^x, where ‘b’ is a positive real number, and ‘x’ is any real number.

In these functions, the variable ‘x’ is the exponent, while ‘b’ is the base. If ‘b’ is greater than 1, we have exponential growth; if ‘b’ is between 0 and 1, we have exponential decay.

### Graphing Exponential Functions

The procedure for graphing an exponential function is as follows:

1. Identify key characteristics, such as the base ‘b’ and whether the function represents exponential growth or decay.
2. Plot the y-intercept (when x = 0, y = 1 for f(x) = b^x).
3. Plot a few more points to outline the shape of the graph.
4. Sketch the graph. Note that the graph will always pass through (0, 1) and never touch the x-axis.

figure

Graph of an Exponential Function

### Practice

Consider the function f(x) = 2^x:

1. The base is 2, indicating exponential growth.
2. The y-intercept is at (0, 1). Plot this point.
3. Plot a few more points. For instance, for x = 1, f(x) = 2^1 = 2, and for x = -1, f(x) = 2^-1 = 0.5.
4. Draw the graph, ensuring it passes through the plotted points and never touches the x-axis.

You’ve successfully graphed an exponential function! As our course progresses, we’ll delve deeper into other types of functions and their graphing techniques. Stay tuned!

[Next: Logarithmic Functions and Their Graphs]

## Part 5: Logarithmic Functions and Their Graphs

Continuing our SAT preparation course at MKSprep, Kathmandu, Nepal, we’re now moving on to logarithmic functions. These functions, which are the inverse of exponential functions, play a vital role in many mathematical concepts.

### Understanding Logarithmic Functions

A logarithmic function is defined as f(x) = log_b(x), where ‘b’ is the base of the logarithm (a positive real number not equal to 1), and ‘x’ is the argument of the logarithm (a positive real number).

The logarithm log_b(x) is the exponent to which ‘b’ must be raised to obtain ‘x’. In simpler terms, if y = log_b(x), then b^y = x.

### Graphing Logarithmic Functions

To graph a logarithmic function, follow these steps:

1. Identify the base of the logarithm and the domain of the function. The domain of any logarithmic function is (0, ∞).
2. Plot the y-intercept if it exists. The basic logarithmic function, f(x) = log_b(x), does not cross the y-axis, but shifted or scaled versions might.
3. Plot a few more points by choosing values for ‘x’ and calculating ‘y’.
4. Connect the points to form the graph, which should be a curve that gets closer and closer to the y-axis but never quite touches or crosses it.

figure

Graph of a Logarithmic Function

### Practice

Consider the function f(x) = log_2(x):

1. The base of the logarithm is 2.
2. This function doesn’t cross the y-axis. It approaches the y-axis but never crosses it.
3. Choose some values for ‘x’ and calculate ‘y’. For instance, for x = 1, y = log_2(1) = 0, and for x = 2, y = log_2(2) = 1.
4. Plot these points and draw the graph, ensuring it approaches but does not touch the y-axis.

Great job! You’ve graphed a logarithmic function! In the next part, we’ll study trigonometric functions and their graphs.

[Next: Trigonometric Functions and Their Graphs]

## Part 6: Trigonometric Functions and Their Graphs

Our journey through the SAT preparation course at MKSprep, Kathmandu, Nepal, now takes us into the world of trigonometry. In this part, we’ll understand trigonometric functions and how to graph them.

### Understanding Trigonometric Functions

Trigonometric functions, including sine (sin), cosine (cos), and tangent (tan), are fundamental in the study of triangles, waves, oscillations, and periodic phenomena in general. These functions are defined using the ratios of the sides of a right triangle.

### Graphing Trigonometric Functions

Let’s focus on the sine and cosine functions, as they are the most basic trigonometric functions and others can be understood in terms of them.

To graph sine and cosine functions, remember:

1. The domain is all real numbers.
2. The range is [-1, 1].
3. They are periodic functions with a period of 2π (or 360° if you’re working in degrees).

figure

Graph of Sine and Cosine Functions

### Practice

Consider the function f(x) = sin(x):

1. Identify the amplitude (maximum height of the wave), which is 1 for the basic sine function.
2. Identify the period of the function. For the basic sine function, this is 2π.
3. Sketch the graph. At x = 0, sin(x) = 0. At x = π/2, sin(x) = 1. At x = π, sin(x) = 0. At x = 3π/2, sin(x) = -1. At x = 2π, sin(x) = 0. These five points (0, 0), (π/2, 1), (π, 0), (3π/2, -1), and (2π, 0) are enough to sketch one period of the function. The graph will repeat this pattern every 2π.

Congratulations on graphing a sine function! In our next part, we will delve deeper into other types of functions and their graphing techniques. Stay tuned!

[Next: Rational Functions and Their Graphs]

## Part 7: Rational Functions and Their Graphs

As we advance further into our SAT preparation course at MKSprep in Kathmandu, Nepal, our attention now turns to rational functions. These functions, involving ratios of polynomials, feature prominently in various areas of mathematics.

### Understanding Rational Functions

A rational function is a function that can be written as the ratio of two polynomials. The general form of a rational function is f(x) = p(x)/q(x), where p(x) and q(x) are polynomials, and q(x) ≠ 0.

The graph of a rational function often includes vertical asymptotes, horizontal asymptotes, or oblique asymptotes, depending on the degrees of the polynomials.

### Graphing Rational Functions

To graph a rational function, follow these steps:

1. Identify any vertical asymptotes by finding the values of ‘x’ that make the denominator zero.
2. Identify the horizontal asymptote (if one exists) by examining the degrees of the polynomials.
3. Calculate and plot a few points on either side of the vertical asymptotes.
4. Sketch the curve, paying attention to the asymptotic behavior.

figure

Graph of a Rational Function

### Practice

Consider the function f(x) = 1/(x-2):

1. The vertical asymptote is at x = 2, with zero denominator.
2. The horizontal asymptote is y = 0 because the degree of the polynomial in the denominator is greater than that in the numerator.
3. Plot a few points, for example, when x = 0, f(x) = -1/2, and when x = 1, f(x) = -1.
4. Sketch the graph, ensuring the curve approaches but does not touch the asymptotes.

Well done! You’ve graphed a rational function. In the next part, we’ll take a look at absolute value functions and how to graph them.

[Next: Absolute Value Functions and Their Graphs]

## Part 8: Absolute Value Functions and Their Graphs

As we wrap up this section of our SAT preparation course at MKSprep in Kathmandu, Nepal, we’ll turn our focus to absolute value functions. These are important functions in mathematics, as they express the “magnitude” or “distance” without regard to direction.

### Understanding Absolute Value Functions

An absolute value function is a function that contains an algebraic expression within absolute value symbols. The general form of an absolute value function is f(x) = |x|. The absolute value of a number is its distance from zero on the number line, and it is always non-negative.

### Graphing Absolute Value Functions

The graph of the basic absolute value function f(x) = |x| is a V-shaped graph that opens upwards. This graph is symmetric with respect to the y-axis.

figure

Graph of an Absolute Value Function

### Practice

Consider the function f(x) = |x|:

1. Identify the vertex. The vertex is at the basic absolute value function’s origin (0, 0).
2. The graph opens upward, meaning it increases to both the left and right of the vertex.
3. Plot a few points. For example, when x = -1 or 1, f(x) = 1; when x = -2 or 2, f(x) = 2.
4. Sketch the graph. The graph will form a V shape, with the vertex at the origin.

Congratulations! You’ve graphed an absolute value function. We’ve covered the fundamental function types and their graphical representations with this. Stay tuned for more math topics in our SAT preparation course!

[Next: Systems of Equations and Inequalities]