Category: Math 11

Week 9 – Pre Calc 11 – Predicting the vertex

Aidan on

This week we learned how to predict the vertex of an equation without graphing it. We have not yet learned the equation to finding the vertex, but predicting the vertex lets us use what we know about quadratic equations without havinv to usw graphing technology. I chose this subjet because I find it to be straight forward and logical. This understanding is important because being able to indentify the vertex is a skill that will be used in math as we continue to develop our skillset.

Lets look at examples a and b that have been handwritten.

We can verify these solutions using graphing techonology to ensure we are correct.

Example C

Verify with desmos

Glossary :

Vertex – the maximum or minimum point on the equation’s parabola.

Parent function – x^{2}

Vertical translations – how far up or down along the y axis the vertex will be.

Horizontal translations – how far right or left along the x axis the equation will be.

Week 8 – Pre Calc 11 – Identifying whether a table of values represents a linear or quadratic function

Aidan on

This week in math we learned the begging steps of graphing a quadratic equation, but before we learned how to do that we learned how to indentify if a function is linear, quadratic, or neither using a table of values. I chose this subject because I like having a visual, such as a table of values, to see what my function would look like. This understanding is important because knowing the difference between a linear equation, which we dealt with in earlier years, and a quadratic equation, which we are dealing with now and will continue to reappear in later math, is a fundemental skill and comprehension to have. For reference, lets look at a linear function on a graph and a quadratic function on a graph.

Linear :

Key Features of Linear Function Graphs (Sample Questions)

Quadratic :

Quadratic Functions and Their Graphs

Example a.)

 

Step 1. My first step is to observe the pattern of x values in my table of values, notice how they are decreasing by 1. Therefore, the difference between the y values may determine the behaviour of the ordered pairs.

Step 2. Find the difference between the first two terms. -2 – (-3) = 1. Repeat this process to find the first difference. You should get 1, 2, 4, and 8. Notice how the first difference is not constant. This indicates we are NOT evaluating a linear function.

Step 3. We can now find the 2nd difference. Use the values of the first term to find the second difference. The first value is 2-(1) = 1. Repeat this process and get 1, 2 and 4. Notice the second difference is NOT constant. this means that it is NEITHER a linear function or quadratic.

Example b.)

Step 1. Notice that the x values are increasing by 2 units. The difference between the y values may detrmine the behaviours of the ordered pairs.

Step 2. Find the first difference. To get our first value, find the difference between 5 and 0; 0 -(5) = -5. If you continue this process for the rest of the values you get -5,-7,-9,-11. Notice how the first difference is NOT constant meaning that it is not a linear function.

Step 3. Find the second difference. Use the values of the first difference, and find the difference between each value. -7 – (-5) = -2. If you continue this process you get -2, -2, -2. The second difference IS constant meaning it is a quadratic function.

Glossary :

Linear function :a function whose graph is a straight line, that is, a polynomial function of degree zero or one.

Quadratic function : the polynomial function defined by a quadratic polynomial.
Photo links :

1,700 × 987

1,640 × 1,638

 

Week 2 – Pre Calc 11 – Writing powers as radicals

Aidan on

This week in math we learned how to write powers as radicals in which we can evaluate or write them as a mixed radicals. The powers that we’re dealing with in this unit are written in fractional and decimal form which at first can make things look confusing. I chose this topic because evaluating fractional powers is a concept that I was previously unfamiliar with, however I have grown to find the topic fascinating and and a very logical straight forward process. This skill and understanding will be important to know for furthering our understanding of powers and roots.

To start, let’s learn how to convert an exponent that is in decimal form into fractional form (it makes writing powers as radicals much easier).

Example 1.

Step 1. Convert the exponent into fractional form. To do this, you need to consider place value; the denominator of the fraction will be the place value. The digits of the decimal will equal the numerator. In this case we can write the numerator 1.5 as 15. The denominator is written as 10 because the 5 is in the tenths place. The new fraction can now be written as \frac{15}{10}. Next we simplify the fraction by the GCF to clarify communication and to avoid misunderstandings. In this case the highest value that 15 and 10 can be divided by is 5; the new fraction is now \frac{3}{2}.

Step 2. We now convert the base of our power (0.25) into fractional form. Similar to converting the power, 0.25 can be written as \frac{25}{100} because the place value is in the 100th. We must now reduce this fraction once again to reduce the odds of any other misunderstandings. The GCF is 25 because it can evenly divided into 100 and our new fraction in its simplist form is \frac{1}{4}.

Step 3. Lets put together our simplified exponent and base to get the power in fractional, simplfied form. The new power is 1/4^{3/2}

Step 4. In order to convert our new power 1/4^{3/2} into a radical we must recognize that the denomator of our fractional exponent is the index, and the numerator is written as the exponent  in radical form.

Step 5. The method I find to be most efficent is to evaluate starting with the index before I distribute the exponent. Knowing this, \sqrt{1} = 1 and \sqrt{4} = 2, because as we have learned a factor of a number that, when multiplied by itself, gives the original number; 2×2 gives our original number,4. Now we can distribute the exponent 3 using in relation to the power law.

Step 6. In other words, the exponent 3 means that there is 3 copies being made of \frac{1}{2}. 1 x 1 x 1 = 1 and 2 x 2 x 2 = 8. The final simplified answer is 0.25^{1.5} = \frac{1}{8}.

Example 2.

In this example, you will find that it cannot be fully evalauted, so instead we can convert it into a mixed radical.

Step 1. Just like in the first example, we can turn the power into radical form by using the denominator of our fractional exponent as the root and the numerator as the exponent. This time there is no decimals to convert. However, upon writing our power in decimal form we can immediately see that (-16) is not a perfect square so we must distribute the exponent 2 to (-16) instead.

Step 2. Distributing the 2 to (-16) affects everything within the brackets; in other terms it means there is two copies of (-16). Two negatives = a positive, so we can look at this expression as being the same as positive 16^{2}. I solved this equation by using long multiplication; this removes the need for a calculator. 16^{2} = 256.

Step 3. We also need to recognize that \sqrt[3]{256} is also not a perfect cube, so we must convert it into a mixed radical to further evaluate it. Firstly, we need to break down 256 using prime factorization. Because we are evaluating a cube root, we can circle and group together each set of 3 of the same value. In this case I circled 2 groups of three 2s. Then, we can move these twos into the coefficient position of the radical. We only need to take one 2 from each group, the rest cancel out. As our coefficient place we now have 2 x 2 or 4. The prime factors that are left, will be used as our radicand; this is 2 and 2 (2×2) which gives us 4 as our radicand. We put everything back into exponential form and we get 4^{3} \sqrt{256}.

Glossary

GCF (greatest common factor) – The highest number that divides exactly into two or more numbers.

Place value – The system in which the position of a digit in a number determines its value.

Power – An expression that represents repeated multiplication of the same factor

Power Law – exponent is being distributed to the value within brackets, multiplying with other exponents.

Prime factorization – a process of writing all numbers as a product of primes.

Prime number –  a whole number greater than 1 whose only factors are 1 and itself.