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Questions and Answers
Consider the geometrical pattern: Term 1 has 5 sides, Term 2 has 9 sides, Term 3 has 13 sides, Term 4 has 17 sides. If this pattern continues, how many sides would Term 7 have?
Consider the geometrical pattern: Term 1 has 5 sides, Term 2 has 9 sides, Term 3 has 13 sides, Term 4 has 17 sides. If this pattern continues, how many sides would Term 7 have?
- 21
- 33
- 25
- 29 (correct)
Consider the geometrical pattern: Term 1 has 5 sides, Term 2 has 9 sides, Term 3 has 13 sides, Term 4 has 17 sides. Which general term can be used to calculate any term's value?
Consider the geometrical pattern: Term 1 has 5 sides, Term 2 has 9 sides, Term 3 has 13 sides, Term 4 has 17 sides. Which general term can be used to calculate any term's value?
- $T_n = 5n - 1$
- $T_n = 3n + 2$
- $T_n = 4n + 1$ (correct)
- $T_n = 2n + 3$
A number pattern starts with 5, 9, 13, 17,... If 'n' represents the term number, and the term values are labeled 'v', what does 'v' represent in the context of graphing a function?
A number pattern starts with 5, 9, 13, 17,... If 'n' represents the term number, and the term values are labeled 'v', what does 'v' represent in the context of graphing a function?
- The change in x
- The y-value or output (correct)
- The x-value or input
- The constant difference
In a graph, what does the y-axis generally represent?
In a graph, what does the y-axis generally represent?
In the context of a graph displayed on the Cartesian plane, which of the following statements accurately describes the x-axis?
In the context of a graph displayed on the Cartesian plane, which of the following statements accurately describes the x-axis?
What characteristic defines all the graphs shown?
What characteristic defines all the graphs shown?
Given that the graph represents number patterns, which of the following can be inferred?
Given that the graph represents number patterns, which of the following can be inferred?
If a linear graph goes through the origin and represents a direct proportion between marbles and sweets, what does this indicate?
If a linear graph goes through the origin and represents a direct proportion between marbles and sweets, what does this indicate?
How are ascending graphs often used in data analysis?
How are ascending graphs often used in data analysis?
Which of the following transformations, when applied to the basic parabola $y = x^2$, results in the graph of $y = (x + 2)^2 - 4$?
Which of the following transformations, when applied to the basic parabola $y = x^2$, results in the graph of $y = (x + 2)^2 - 4$?
In a quadratic equation, what does the point where the equation equals 0 (i.e., y = 0) on a graph indicate?
In a quadratic equation, what does the point where the equation equals 0 (i.e., y = 0) on a graph indicate?
If the parabola is in its minimum point where it stops decreasing and makes a turn up again, what would you call that point?
If the parabola is in its minimum point where it stops decreasing and makes a turn up again, what would you call that point?
For a parabola described by the equation $y = a(x - h)^2 + k$, how does the value of 'a' affect the graph?
For a parabola described by the equation $y = a(x - h)^2 + k$, how does the value of 'a' affect the graph?
What is the vertex of the parabola represented by the equation $y = (x + 3)^2 - 4$?
What is the vertex of the parabola represented by the equation $y = (x + 3)^2 - 4$?
Why is the minimum value of the parabola $y = a(x - h)^2 + k$ the coefficient 'k' of the equation?
Why is the minimum value of the parabola $y = a(x - h)^2 + k$ the coefficient 'k' of the equation?
If a quadratic equation is expressed in the form $(px + q)(x+d) =0$, what does this tell us about solving the equation?
If a quadratic equation is expressed in the form $(px + q)(x+d) =0$, what does this tell us about solving the equation?
Solve this equation: $(p + 2)(p + 3) = 0$. What are the values of p?
Solve this equation: $(p + 2)(p + 3) = 0$. What are the values of p?
If a linear equation has one value for the variable, but a quadratic equation has two, what does this imply about their graphical representations?
If a linear equation has one value for the variable, but a quadratic equation has two, what does this imply about their graphical representations?
What do the solutions of a quadratic equation $x^2 + 7x + 6 = 0$ represent graphically?
What do the solutions of a quadratic equation $x^2 + 7x + 6 = 0$ represent graphically?
Flashcards
Graphing functions
Graphing functions
A function represented visually on the Cartesian plane.
y-axis
y-axis
The vertical axis, marked with positive numbers upwards from the origin and negative numbers downwards.
x-axis
x-axis
The horizontal axis, marked with positive numbers to the right and negative numbers to the left from the origin.
Linear Equations
Linear Equations
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Quadratic Equations
Quadratic Equations
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Study Notes
- In $ax + 2 = 0$ if either $(x + b) = 0$ or $(x + c) = 0$, then either $x + b = 0$ or $x + c = 0$.
- Linear equations can have only one value for the variable.
- Quadratic equations can have two possible values for the variable.
- Activity 1.4.3(b) follows Activity 1.4.3(a), factoring quadratic expressions.
- Solve for $x$ in the equations below and explain each step.
- Activities follow methods to solve.
Graphing Functions
- Graphing functions is one of the four major processes in algebra.
- Verbal description and extension of visual (geometric) patterns is how Learners start very early in school.
- Single out countable properties of geometrical patterns and learn values attached to each term in the patterns.
- Terms such are numbers and values allow for finding the rule for finding nth term.
- The rule can be represented in a flow chart or table.
- The rule can be applied to any term of the pattern to find its value.
- Only a basic idea of graphing and representing functions are included, technical details are not included.
- The y axis is the vertical axis, marked in positive numbers from the origin, zero point, and in negative numbers downwards.
- The x axis is the horizontal axis, marked in positive numbers from the origin to the right and in negative numbers from the origin to the left.
Analyzing Geometrical Patterns
- Term 1 starts with a shape with five sides.
- Term 2 involves the first five-sided shape is added.
- The added shape shares a side with the first shape, therefore only four sides are newly added.
- The result is a shape with nine sides.
- Every time another four sided shape is added where another four-sided shape is added, it shares a side with the previous, so only four sides are added.
- The number of sides increases by four each time a shape is added.
- The number pattern is 5, 9, 13, 17....
- The constant difference and first terms are important elements of the rule.
- The first term is 5 and the constant difference is 4.
- The "jumps" are by 4 and therefore one more than the constant difference to start with.
- The general term for the pattern that can be used to calculate any term's value is $Tn$.
- The general term is $Tn = 4n + 1$.
- If we have the general term, any term number $(n)$ can be inserted and can find the value of that term.
Analyzing a Graph
- The graph is to express the number pattern 1, 3, 5.
- In the linear equation: $kx + 1$ or $f(x) = kx + 1$.
- Inspect the graph and then answer the following questions.
- Describe the rate of change from the graph.
- Linear graph goes through the origin and represents a direct proportion between two entities.
- On succeeding graphs when use on increase or decreasing scales reference the quadratic graph of ax² + bx + c
Quadratic Equations
- At the point where the equation = 0 or y = 0, the variable has two values.
- Find where the parabola intercepts with the x axis.
- The parabola is a perfect symmetry in line trigonometry passes through the axis at the middle of $-3$ and $-4$ and also through the minimum point where the parabola stops dropping and makes it turn-up again.
- Y = 6.25 (this can also say the minimum value of this parabola is y = at (x=2.5).
- This happens whenever $6.25$ replaces with numbers otherwise find.
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