This week we started our trigonometry unit. This was one of the highlights of math ten for me. This week we reviewed the acronym of SOH CAH TOA, this helps me remember which formula to use when finding a side of a right triangle.
sine = opposite/hypoteneuse
cosine = adjacent/hypoteneuse
tangent = opposite/adjacent
We reviewed how to find theta or an unknown angle using the inverse sine, cosine or tangent. And isolating our unknown variable like we know how to do from the solving unit. We then looked at drawing right angle triangles in circles. We would connect our 4 quadrants of a graph with a circle and starting from the center we would draw our reference angle out to the edge of the circle becoming our radius. And at the tip of our reference angle is our reference point written as a coordinate : (#,#). The first number shows the x axis or the adjacent side and the second number shows the y axis or the opposite side. This reference angle formula can be known now as:
sine = y/radius
cosine = x/radius
tangent = y/x
This is how we can look at triangles in quadrant one. Using negative numbers we can assess triangles in all quadrants of the graph. We learned a new acronym for looking at the signs of numbers in each quadrant: All Students Take Calculus, ASTC. This means that in quadrant one all equations will be positive. In quadrant 2 sine will be positive, in quadrant 3 tangent will be positive and in quadrant 4 cosine will be positive.
This week we learned about adding and subtracting rational fractions. This skill was a little trickier than multiplying and dividing. When subtracting I would usually change the signs of each value in the second numerator so that every question would be addition. Then I would find a common denominator of the two equations. I would multiply the numerator by what is missing in the denominator. This would simplify the equation and when solving, get rid of the denominator. Then I can simplify the coefficient.
This week we learned multiplying and dividing radical fractions. Our first step is to factor and then find the non-permissible values. Remember factoring 1-2-3: take out something in common, difference of squares and trinomial factoring. When multiplying fractions, we can take out whats in common right away after factoring. When dividing fractions, we can multiply by the recipricol of the second fraction. So when finding the NP values we need to look at the denominators before and after we flip the fractions.
We made a stop motion video that describes Newton’s Laws and how they can apply to driving, specifically young drivers.
Newton’s Laws Video
This week we hit the halfway point in our graphing quadraticfunctions unit. We also had a chance to review all of the previous chapters for the midterm coming up next week. First we learned to analyze quadratic function. Or graphing word problems.
The best strategy I have for this one is to use the information we know and put it into an equation with an . Then I can use strategies for graphing by changing my function between standard, general and factored forms.
The kids at the cafeteria sell french fries for $2. At this price they sell 100 fries a day. They experimented and found that every $1 increase in price they would sell 20 fewer fries. What price of fries will maximize the revenue of the cafeteria?
(price)(#sold) = revenue
(2 + 1y) x (100 – 20y) = revenue <– Find x intercepts and axis of symmetry.
y = 2, y = 5
2+5/2 = 7/2 <– input for y
(2 + 7/2)
11/2 = price
This Monday, we learned about standard form or vertex form. This is our function expressed in a format where we can easily determine the vertex, shape, stretch or compression and more about our function. Standard form looks like this: y = + q
Where a shows us if the parabola is opening up (being positive) or down (being negative). If a is greater than one; the parabola is stretched, if a is less than one; the parabola is compressed. The p does not include the negative sign. So the p is generally the opposite sign of what it appears. P represents the horizontal translation, whether the vertex will move left or right. The q in the function indicates the vertical translation of our parabola. P and Q together show us our vertex! So there is more than enough information we can gather from standard form.
y = + 2
a = -1 The parabola is facing down and congruent to our parent function. p = 4 The vertex is four units to the right. q = 2 The vertex is up two units. The vertex is (4, 2)
Standard form is super helpful! When given a point of our parabola, we can determine many aspects of our function. If our vertex is (3, 5) and it passes through the point (1, 2) we can find out everything about the function. We can plug in our values into the x and y places.
We take standard form: y = + q. Then, we insert what we know: 2 = + 5. Then we can solve to find our a, stretch or compression.
2 = a(4) + 5 -3 = a(4) a = -3/4
After forming a good understanding of solving quadratic equations, this week we learned how to graph quadratics.
First we reviewed some graphing methods from grade 10. Like using y=mx+b form and making a t-chart to graph some points. Quadratic formulas do not make a linear graph like we focused on last year. Quadratics will make a shape called a parabola. This is kind of like a U shape. So that if we were to draw a line anywhere across our points there would be two solutions, unless the line crosses the vertex. Here is y = exponent graphed:
This U shape is a parabola. We discussed 7 things that we can determine about this graph:
- The vertex – This is the middle of the parabola where the curve changes. The vertex is the most important part of the graph. It can show us where the minimum or the maximum range of our graph.
- The line of symmetry – This is a line that can be drawn from our vertex up the center of the parabola.
- The x-intercept – The is where our parabola crosses the x-axis. In this case, it is our vertex. Usually, there will be two x-intercepts in a quadratic function.
- The y-intercept – Where our parabola crosses the y-axis.
- Domain – The domain is all of our restrictions for our x values. In this case: xER
- Range – The range is all of our restrictions for our y values. In this case: y≥0
- Minimum and maximum – This is our vertex. Our minimum point on our graph is zero, at our vertex. If the parabola was a reflection of the example, our vertex would show our maximum value.
Here is my reflection on my studio arts Halloween lantern.
This week we learned another way to solve quadratic equations. Last week and this week we reviewed three methods: square roots, factoring or zero product law and complete the square. On Wednesday we talked about when to use which strategy. When there is one variable: , we can simply solve using square roots. We can factor when the expression is in our trinomial pattern: + x + #. And finally we complete the square when the equation is not factorable.
We now have another method to solve for x: quadratic formula. This formula comes from complete the square method and can be used for any quadratic equation.
The quadratic formula is : x =
If we were to look at the equation + 6x + 3
We know that this does not factor so we can use complete the square or the quadratic formula. My first step is to write down what I know:
a = 2 b = 6 c = 3
Now we can input these numbers into the quadratic formula and solve for x. The nice thing about the quadratic formula is that x is already isolated.