diff --git a/Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md b/Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md index 26631ce..6e73166 100644 --- a/Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md +++ b/Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md @@ -1,7 +1,8 @@ # Unit 3: Quadratic Functions ## Definitions - - `Linear relation`: A relation with a constant rate of change + - `Linear relation`: A relation with a constant rate of change, or with a constant first difference + - `Quadratic relation`: A relation where the 2nd differences are constant - `Non-linear relation`: A relation with a variable rate of change - `Degree` of a polynomial: Highest exponent that appears in any term of a polynomial - `a-value`: Step property @@ -15,8 +16,114 @@ - `Function`: A relation in which no y-coordinates share an x-coordinate (e.g., circles are not functions) ## Parabolas +A standard graph of a parabola $`y=x^2`$ would look something like this: + + + +There are several things you should know about a parabola: +- X-intercepts +- Y-intercepts +- Vertex +- Optimal value +- Axis of Symmetry +- Direction Of Opening +- Step Property + + +### X-intercepts +- These are the zeroes of the quadratic function, or the solutions you found when solving for a quadratic function in factored form. +- These are the values of $`x`$ where $`y=0`$. +- By using factored form, we can easily see that an equation with factored form of $`y=a(x-r)(x-s)`$ has its x-intercepts at $`(r, 0), (s, 0)`$. + +### Y-intercepts +- These are the values of $`y`$ where $`x=0`$. + +### Vertex +- This highest/lowest value of $`y`$ that the parabola takes. +- This point tells us alot of things, including the `axis of symmetry` and the `maximum/minimum/optimal` value. +- With vertex form in $`y=a(x-h)^2 + k`$, we know that the vertex is at $`(h, k)`$. + +### Optimal value +- This is basically the $`y`$ value of the vertex, and is useful for maximum/minimum word problems +- If the parabola is opening upwards, this is the minimum value. If the parabola is opening downards, this is the maximum value + +### Axis of Symmetry +- A vertical line of symmetry for the parabola. +- This can be determined in many ways: + - Using factored form $`y=a(x-r)(x-s)`$, the axis of symmetry is at $`\dfrac{r+s}{2}`$ + - Using standard form $`y=ax^2+bx+c`$, the axis of symmetry is at $`\dfrac{-b}{2a}`$ + - Using vertex form, $`y=a(x-h)^2+k`$, the axis of symmetry is simply the $`x`$ coordinate of the vertex, so $`h`$. + +### Direction Of Opening +- To put it bluntly, the parabola opens upward if the $`a`$ value is positive, and downwards if the $`a`$ value is negative. +- Think of a postive $`a`$ value as a **happy face**, and a negative $`a`$ value as a **sad face**. + +### Step Property +- This is a property that can be used to quickly graph a quadratic function. +- The step property for an $`a`$ value of $`1`$ is $`1, 3, 5, 7, \cdots 2n+1`$ for any $`n \ge 1`$. The step property for any other a value is $`1a, 3a, 5a, 7a, \cdots a(2n+1)`$ for any $`n \ge 1`$. +- The step property tells us the difference of values between each point starting from the vertex, meaing if the vertex is at $`(x, y)`$,
the next two points would be $`(x-1, y+n), (x+1, y+n)`$, where $`n`$ is the $`n^{th}`$ step property number. + +## Functions +`Function`: A relation in which no y-coordinates share an x-coordinate (e.g., circles are not functions) + +### Vertical Line Test (VLT) +We can easily tell if a relation is a function by using the vertical line test. If a single straight line of $`x=n`$ for any $`n`$ has more than $`1`$ point on the function, then that relation is not a function. Bluntly, a function cannot have a vertical straightline touching any $`2`$ of its points. + +### Ways of Representing Functions + +#### 1. Table Of Values +|$`x`$|$`y`$| +|:----|:----| +|$`2`$|$`5`$| +|$`5`$|$`7`$| +|$`6`$|$`5`$| + +#### 2. Coordinates in a Set +$`f = \{(-2, 1), (0, 1), (3, 1), (4, 1), (7, 1)\}`$ + +#### 3. Graph + + +#### 4. Mapping (Bubble Diagrams) + + +#### 5. Equation +$`y=x^2-5`$ + +#### 6. Function Notation +$`f(x) = x^2 - 5`$ + +### Domain And Range +- A `domain` is the set of $`x`$-values, and the `range` is the set of $`y`$-values. +- To represent domain and range, we use **set notation** to represnt it or simply by listing the $`x`$ and $`y`$ values. +- We use listing method for representing points, such as: + - + - $`D = \{-3, -1, 0, 1, 2, 2.5 \}`$ + - $`R = \{-2, -1, 0.5, 3, 3, 5 \}`$ + - Make sure the values or sorted in order. +- We use **set notation** or **interval notation** to represent a continous graph, such as: + - + - $`D = \{x \mid x \in \mathbb{R} \}`$ + - $`R = \{y \mid y \ge 2, y \in \mathbb{R} \}`$ ## Transformations +There are 3 main types of transformations for a quadratic function. + +### Vertical Translation +- When we graph the quadratic relation $`y=x^2+k`$, the vertex of the parabola has coordinates $`(0, k)`$ +- When $`k \gt 0`$, the graph of $`y=x^2`$ is **vertically** translated **up** $`\mid k \mid`$ units. +- When $`k \le 0`$, the graph of $`y=x^2`$ is **vertically** translated **down** $`\mid k \mid`$ units. + +### Horizontal Translation +- When we graph the quadratic relation, $`y=(x-h)`$ the vertex of the parabola has coordinates $`(h, 0)`$ +- When $`h \gt 0`$, the graph of $`y=x^2`$ is **horizontally** translated **left** $`\mid k \mid`$ units. +- When $`h \le 0`$, the graph of $`y=x^2`$ is **horizontally** translated **right** $`\mid k \mid`$ units. + +### Vertical Stretch/Compression +- If $`a \lt 0 \rightarrow`$, the graph is **reflected** over the x-axis. +- If $`\mid a \mid gt 1 \rightarrow`$ vertical stretch.expansion by a **factor of** $`\mid a \mid`$. +- If $`0 \lt \mid a \mid \lt 1 \rightarrow`$ vertical compression by a factor of $`\mid a \mid`$ +- The step property also gets affected. ## Forms of Quadratic Functions