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HSC Mathematics: Exam Prep & Syllabus35 chapters | 250 lessons | 18 flashcard sets

Instructor:
*Anna Teper-Dillard*

I was born and raised in Poland. Before I ended up settling in Chicago, I traveled through Europe and Asia. I have been to China, Mongolia, Russia, Italy, Spain, Germany, Czech Republic and England. In my free time, I love reading, and being physically active. I ski, bike, swim and run long distance. I finished the Chicago Marathon, the Fox Valley Marathon, a couple of Half-Marathons and many shorter races. In the summer of 2017, I will be participating in 70.3 miles Iron Man triathlon in Michigan. Currently, I live in Woodrigde, IL with my husband and three children. I have two master degrees, one in Secondary Education and the other in Science of Mathematics. My philosophy on teaching and learning :) I am a long-life learner. Discovering new knowledge (in any subject) is fascinating and rewards us with a broader and more informative point of view. Being exposed to wider range of experiences gives us an edge in life. Although grades we earn throughout our education do not communicate the whole story about who we are, they are good indicators of many attributes and skills we have accomplished: perseverance, fortitude, hard, consistent work, collaboration, reflection on our own mistakes and yes, humbleness.

In this lesson you will learn about the characteristics of a square root function and how to graph it. You will also examine in detail the effect the square root sign has on the independent and dependent quantities.

A **square root function** is a function that has the radical (square root sign) and the independent quantity *x* is the radicand (value under the square root sign). The square root function in its basic form has the following equation:

However, to truly understand the behavior of a square root function, let's look at the basic linear function:

f(x) = x

Based on the equation, every *y* value (the dependent quantity) will be exactly the same as the independent quantity, the *x* value. In other words, what you choose for *x* will output the same value for *y*.

Now, if we changed the equation by taking the square root of the right side, the function would become the basic square root function. You might ask however, what is taking the square root? When we square a number, we really just multiply it by itself.

**Square root** is the inverse operation of squaring a number. In other words, we are trying to find a factor that was multiplied by itself to result in your original value.

Let's look at the square root function now that we have all the basics down!

As we expected, the change in the equation resulted in the change of the values in the table and the behavior of the graph. What we should notice first is the *undefined* message in the table of values. Depending on the calculator you are using to graph it, you might see *error* instead. The undefined value comes from the operation of square rooting a negative number. There is no negative number that can be multiplied by itself and result in a negative number! Only a positive number can be a result of two negatives. Here are a few examples of this operation:

That is why in the table of values we only see the square root of positive *x* values which results in the positive *y* values. When we look at the graph again, we see that distinct beginning point, where it is possible to square root a number, in this case zero.

**Domain** is the set of all *x* independent values for which the function f(x) exists or is defined.

**Range** is the set of all *y* dependent values that will result from substituting all *x* values (domain) into the function.

In a square root function, both domain and range are restricted based on the concept that we can only square root positive numbers.

Let's look at a different example of domain and range using the square root function.

The graph has been shifted and so did the values. Notice how, zero is no longer a valid *x* value for this function.

Both domain and range will be affected by the changes applied to the basic square root function. As you can see in the table and the graph, the smallest *x* value is 2, and the smallest *y* value is 3.

**Horizontal translation** is a shift of the graph and all its values either to the left or right. The change will occur if we add or subtract a number from *x* under the radical sign.

The g(x) has shifted 2 units to the right, not left! Instead of points (0, 0), (1, 1), and (2, 1.41) on the f(x) graph, we have (2, 0), (3, 1) and (4, 1.41) on the g(x) graph. Notice how all the *x* values were increased by 2, while the *y* stayed the same. So essentially, if the number is subtracted from *x* in the equation, the graph and all the *x* values, move to the right.

What can you predict about the behavior of this square root function? Do you think h(x) will translate 4 units to the left or right?

If you said the graph of h(x) will shift 4 units to the left you were correct! This is because the negative number added to the x in the equation caused the graph and all of the x values to shift to the left.

**Vertical Translation** is shifting a graph up or down on the plane of coordinates. The vertical translation occurs when we add or subtract a number outside of the function, in our case, outside of the square root sign.

Let's look at this following change in the equation.

We did add 4, just like in the g(x) example, but there is a major difference to where we performed that addition. It's no longer under the square root sign, it is outside. In this case, the *y* values will be affected by this transformation. If using a calculator to compute the vertical translation of 'x', determine the square root, then add the number 4. For example, the square root of 1 is 1, and 1 + 4=5.

Adding or subtracting a number outside of the square root sign will translate the graph vertically, move all the *y* values according to the algebraic operation indicated by its value. If we add 4, the graph shifts up, if we subtract 4 from the equation, the graph will translate 4 units down. Logical, right?!

If the graph is translated vertically the range will change with that shift. Domain, all *x* values, will stay the same as of the basic square root function.

There are two major types of reflections when it comes to graphs, and they are vertical and horizontal.

**Vertical reflection** is a reflection of the graph over the *x-axis*. We obtain the vertical reflection by multiplying the function by -1.

As you see, the *y* values became negative, because we multiplied the function by -1. Notice, that the *x* values are not affected, they are the same in j(x) as they were in f(x). Therefore, domain is zero to infinity, while range is from negative infinity to zero.

**Horizontal reflection** is a reflection across the *y-axis*. We obtain the horizontal reflection by multiplying the *x* value by -1.

Because we multiplied the input of the function by -1, all *x* values became negative, while the *y* stayed the same. So domain is from negative infinity to zero and range is zero to infinity.

1. Basic square root function takes the square root of independent quantity.

2. Domain and range of a basic square root function are restricted, because the square root of a negative number does not exist. Both domain and range of the basic function are from zero to infinity.

3. Horizontal translation occurs when we add or subtract a number under the square root sign. The operation shifts the graph horizontally to the opposite direction of the sign of the number.

4. Vertical translation occurs when we add or subtract a number outside of the square root sign. The operation shifts the graph vertically according to the sign of the number.

5. Horizontal reflection reflect the graph across the *y-axis*, and it occurs when we multiply the *x* under the square root sign by -1.

6. Vertical reflection reflects the graph across the *x-axis*, and it occurs when we multiply the function outside the square root sign by -1.

7. Domain and range are affected by the transformations.

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HSC Mathematics: Exam Prep & Syllabus35 chapters | 250 lessons | 18 flashcard sets

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