MA19.PRE.28c
Read values of an inverse function from a graph or a table, given that the function has an inverse.
Read values of an inverse function from a graph or a table, given that the function has an inverse.
MA19.PRE.28c
MA19.PRE.28d
Produce an invertible function from a non-invertible function by restricting the domain.
Produce an invertible function from a non-invertible function by restricting the domain.
MA19.PRE.29
Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents. Extend from logarithms with base 2 and 10 to a base of e.
Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents. Extend from logarithms with base 2 and 10 to a base of e.
Unpacked Content
Knowledge
Students know:
- Definition of Logarithm. If bx = y, then logb y = x.
- Restrictions on domain and range of both functions (b > 0, b not equal to 1, y > 0).
- Techniques for creating graphs of exponential and logarithmic functions.
- Situations that can be modeled by exponential and logarithmic functions.
Skills
Students are able to:
- Re-write the equivalent inverse function from an exponential function or logarithm function in various forms (graphs, tables, or equations).
- Model contextual situations using logarithmic and exponential functions.
- Solve exponential and logarithmic equations by isolating the variable.
Understanding
Students understand that:
- Equivalent forms of a function, specifically logarithmic and exponential, may be useful at different times to solve problems.
- Although the algebraic representations look different, the logarithmic form and exponential form of the same relationship are equivalent.
Vocabulary
- Inverse relationship
- Logarithm
- Natural logarithm
MA19.PRE.30
Identify the effect on the graph of replacing $f(x)$ by $f(x) + k$, $k \cdot f(x)$, $f(k \cdot x)$, and $f(x + k)$ for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs. Extend the analysis to include all trigonometric, rational, and general piecewise-defined functions with and without technology.
COS Examples
Example: Describe the sequence of transformations that will relate y=sin(x) and y=2sin(3x).
Identify the effect on the graph of replacing $f(x)$ by $f(x) + k$, $k \cdot f(x)$, $f(k \cdot x)$, and $f(x + k)$ for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs. Extend the analysis to include all trigonometric, rational, and general piecewise-defined functions with and without technology.
COS Examples
Example: Describe the sequence of transformations that will relate y=sin(x) and y=2sin(3x).
Unpacked Content
Knowledge
Students know:
- Graphing techniques of functions.
- Methods of using technology to graph functions.
Skills
Students are able to:
- Accurately graph functions.
- Check conjectures about how a parameter change in a function changes the graph and critique the reasoning of others about such shifts.
- Identify shifts, stretches, or reflections between graphs.
Understanding
Students understand that:
- Graphs of functions may be shifted, stretched, or reflected by adding or multiplying the input or output of a function by a constant value.
Vocabulary
- Horizontal shift
- Vertical shift
- Horizontal stretch/shrink
- Vertical stretch/shrink
- Reflection
- Phase shift
- Amplitude
- Midline
MA19.PRE.31
Graph conic sections from second-degree equations, extending from circles and parabolas to ellipses and hyperbolas, using technology to discover patterns.
Graph conic sections from second-degree equations, extending from circles and parabolas to ellipses and hyperbolas, using technology to discover patterns.
Unpacked Content
Knowledge
Students know:
- Vertex form of a parabola.
- Standard form of a circle.
- Vertex and axis of symmetry of a parabola.
- Completing the square.
- Factoring a quadratic function.
Skills
Students are able to:
- Graph equations of parabolas.
- Graph equations of circles.
- Graph equations of ellipses.
- Calculate eccentricities of ellipses.
- Graph equations of hyperbolas.
- Classify a conic section using its general equation and/or its discriminant.
Understanding
Students understand that:
- A conic section is a graph of an equation of the form Ax2 + Bxy + Cy2 +Dx +Ey + F = 0.
- The only conic sections that are functions are parabolas that open upward or downward, previously learned as quadratic functions and hyperbolas that are written in the form of a rational function.
- Using algebra to manipulate the equation of a conic section, particularly the method of "completing the square"" can be used to determine the parts and properties of its graph
Vocabulary
- Hyperbola
- Ellipse
- Degenerate conic
- Focus (foci)
- Latus rectum (focal distance)
- Major axis (transverse axis)
- Minor axis (conjugate axis)
- Eccentricity
- Asymptote
- Directrix
- Locus
MA19.PRE.31a
Graph conic sections given their standard form.
COS Examples
Example: The graph of $\frac {x^2}{9} + \frac {(y-3)^2}{4} = 1$ will be an ellipse centered at (0,3) with major axis 3 and minor axis 2, while the graph of $frac {x^2}{9} - \frac {(y-3)^2}{4} = 1$ will be a hyperbola centered at (0,3) with asymptotes with slope $
\neq 2/3.$
Graph conic sections given their standard form.
COS Examples
Example: The graph of $\frac {x^2}{9} + \frac {(y-3)^2}{4} = 1$ will be an ellipse centered at (0,3) with major axis 3 and minor axis 2, while the graph of $frac {x^2}{9} - \frac {(y-3)^2}{4} = 1$ will be a hyperbola centered at (0,3) with asymptotes with slope $ \neq 2/3.$
MA19.PRE.31b
Graph conic sections given their standard form.
COS Examples
Example: The graph of $\frac{x^2}{9} + \frac{(y−3)^2}{4} = 1$ will be an ellipse centered at (0,3) with major axis 3 and minor axis 2, while the graph of $\frac{x^2}{9} − \frac{(y−3)^2}{4} = 1$ will be a hyperbola centered at (0,3) with asymptotes with slope $
\neq 2/3.$
Graph conic sections given their standard form.
COS Examples
Example: The graph of $\frac{x^2}{9} + \frac{(y−3)^2}{4} = 1$ will be an ellipse centered at (0,3) with major axis 3 and minor axis 2, while the graph of $\frac{x^2}{9} − \frac{(y−3)^2}{4} = 1$ will be a hyperbola centered at (0,3) with asymptotes with slope $ \neq 2/3.$
MA19.PRE.32
Solve application-based problems involving parametric and polar equations.
Solve application-based problems involving parametric and polar equations.
Unpacked Content
Knowledge
Students know:
- How to model motion using quadratic functions.
- Algebraic manipulation of equations.
Skills
Students are able to:
- Develop parametric and polar equations for a given situation.
- Graph the parametric or polar equations.
- Answer questions in the context of the parametric or polar problem.
Understanding
Students understand that:
- Parametric equations can be used to model and evaluate trajectory of projectiles.
- Parametric equations can be used to model and evaluate range of projectiles.
- Polar equation can be used to model motion for some mechanical systems.
- Which situations parametric and polar modeling is appropriate.
Vocabulary
- Parametric equations
- Parameter
- Parametric curve
- Polar equations
- Rectangular form
- Orientation
MA19.PRE.32a
Graph parametric and polar equations.
Graph parametric and polar equations.
MA19.PRE.32b
Convert parametric and polar equations to rectangular form.
Convert parametric and polar equations to rectangular form.
MA19.PRE.33
Use special triangles to determine geometrically the values of sine, cosine, and tangent for $\frac{\pi}{3}$, $\frac{\pi}{4}$, and $\frac{\pi}{6}$, and use the unit circle to express the values of sine, cosine, and tangent for $\pi - x$, $\pi + x$, and $2\pi - x$ in terms of their values for $x$, where $x$ is any real number.
Use special triangles to determine geometrically the values of sine, cosine, and tangent for $\frac{\pi}{3}$, $\frac{\pi}{4}$, and $\frac{\pi}{6}$, and use the unit circle to express the values of sine, cosine, and tangent for $\pi - x$, $\pi + x$, and $2\pi - x$ in terms of their values for $x$, where $x$ is any real number.
Unpacked Content
Knowledge
Students know:
- The relationship between the lengths of the sides of a 45-45-90 and 30-60-90 triangle.
- The basic trig ratios.
Skills
Students are able to:
- Find the value of sine, cosine, and tangent of π/3, π/4 and π/6 using special triangles.
- Locate an angle in standard position in the unit circle.
- Convert between degrees and radians.
Understanding
Students understand that:
- For an angle in standard position, the point where the terminal ray intersects the unit circle has an x-coordinate which is the value of the cosine of the angle and a y-coordinate which is the value of the sine of the angle.
- Patterns that can be identified on the unit circle allow for application of right triangle trigonometry to angles of all sizes.
Vocabulary
- Special triangles
- Unit circle
MA19.PRE.34
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Unpacked Content
Knowledge
Students know:
- The characteristics of the unit circle.
- The characteristics of even and odd functions.
Skills
Students are able to:
- Transpose coordinates from the unit circle to a graph on the xy-plane.
- Use the unit circle to verify even and odd trig identities.
Understanding
Students understand that:
- Basic trig functions can be classified as either even or odd.
- The repetitive nature of the unit circle creates periodic functions which is displayed in their graphs.
Vocabulary
- Odd and even symmetry
- Periodicity
MA19.PRE.35
Demonstrate that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
Demonstrate that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
Unpacked Content
Knowledge
Students know:
- Characteristics of an always increasing or always decreasing function.
- When a function has an inverse that is also a function.
Skills
Students are able to:
- Identify intervals where the function is increasing or decreasing.
- Identify the domain of a function that will produce an inverse.
- Produce different models of functions and their resulting inverse.
Understanding
Students understand that:
- Due to the periodic nature of trig functions, domains have to be restricted in order to produce a one-to-one relationship.
- While there are many different intervals that are always increasing or always decreasing, there are conventional choices for the restricted domain of the trig functions.
Vocabulary
- Restricting the domain
- Inverse functions
MA19.PRE.36
Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
Unpacked Content
Knowledge
Students know:
- Periodic situations are best modeled by trig functions.
- Solutions that are mathematically possible may not be physically possible in the context of the problem.
- Determine when technology is appropriate to finding a solution.
Skills
Students are able to:
- Solve a trig equation.
- Interpret the meaning of the solution.
- Interpret a domain in a contextual situation.
- Use technology to solve a trig equation.
- Translate a computed solution to an equivalent solution that fits in the physical domain (i.e., If -30 degrees is a solution, then another solution can be 330 degrees).
Understanding
Students understand that:
- There are periodic phenomena in real life that can be modeled by trig functions.
- From the many solutions that result from a periodic function, only some may be logical given the context.
Vocabulary
- Inverse functions
- Trigonometric equations
MA19.PRE.37
Use trigonometric identities to solve problems.
Use trigonometric identities to solve problems.
Unpacked Content
Knowledge
Students know:
- Applications of the Pythagorean Theorem.
- Operations with trigonometric ratios.
- Operations with radians and degrees.
- Even and odd functions.
Skills
Students are able to:
- Use and transform the Pythagorean Identity.
- Simplify trigonometric expressions.
- Verify trigonometric identities.
- Write the sum and difference identities for sine, cosine, and tangent.
- Use sum and difference identities to findexact values of a trig function.
- Derive the double angle and half angleidentities.
Understanding
Students understand that:
- The fundamental identities allow functions to be written in terms of other functions. Then algebraic methods can be applied to simplify expressions or to match it with another expression
- Given the trig values for a pair of angles, identities can be used to find the trig values of the sum or difference of the given angles.
- Given the trig values for an angle, identities can be used to find the trig values for twice and half the angle.
Vocabulary
- Identity
- Even and odd function
- Sine
- Cosine
- Tangent
- Cosecant
- Secant
- Cotangent
- Fundamental Identities
- Reciprocal Identities
- Quotient Identities
- Pythagorean Identities
- Sum and Difference Identities
- Double-Angle Identities
- Half-Angle Identities
MA19.PRE.37a
Use the Pythagorean identity $sin^2 ( heta) + cos^2( heta) = 1$ to derive the other forms of the identity.
COS Examples
Example: $1 + cot^2 ( heta) = csc^2 ( heta)$
Use the Pythagorean identity $sin^2 ( heta) + cos^2( heta) = 1$ to derive the other forms of the identity.
COS Examples
Example: $1 + cot^2 ( heta) = csc^2 ( heta)$
MA19.PRE.37b
Use the Pythagorean identity $\sin^2 (\theta) + \cos^2(\theta) = 1$ to derive the other forms of the identity.
COS Examples
Example: $1 + cot^2 ( \theta) = csc^2 ( \theta)$
Use the Pythagorean identity $\sin^2 (\theta) + \cos^2(\theta) = 1$ to derive the other forms of the identity.
COS Examples
Example: $1 + cot^2 ( \theta) = csc^2 ( \theta)$
MA19.PRE.37c
Use the Pythagorean and double angle identities to prove other simple identities.
Use the Pythagorean and double angle identities to prove other simple identities.
