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Questions and Answers
What are the equations of the asymptotes of the function f(x) = \frac{3}{5} - 2. x - 1?
y = \frac{3}{5} and x = 1
What are the intercepts of the graph of f with the axes?
x-intercept: x = 1, y-intercept: y = \frac{3}{5}
What is the range of y = -f(x)?
All real numbers
Describe the transformation of f to g if g(x) = \frac{-2}{x + 1}.
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What are the coordinates of point C, where the parabola f intersects the x-axis?
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What is the x-coordinate of point B on the parabola f?
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What is the general form of the equation of f?
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What is the equation of the reflection of f in the x-axis?
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What is the maximum value of t(x) if t(x) = 1 - f(x)?
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Is the function f(x) = \frac{1}{3} increasing or decreasing?
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Study Notes
Question 5
- The function f is defined as f(x) = 3/(x-2) - 1.
- Calculate the vertical asymptote by setting the denominator to zero: x - 2 = 0 and solving to find x = 2
- Calculate the horizontal asymptote by considering the degree of the numerator and denominator: degree of numerator (0) < degree of denominator (1), therefore horizontal asymptote is y = 0
- Find the y-intercept by setting x = 0 in the equation and solving for y.
- Find the x-intercept by setting y = 0 in the equation and solving for x.
- Sketch the graph of f, showing the asymptotes and the intercepts.
- To determine the range of y = -f(x), reflect the graph of f in the x-axis and note the y-values that the graph takes on.
- Describe the transformation of f to g, where g(x) = -3/(x+1) - 2, in words. This involves a reflection in the x-axis, a shift of 3 units to the left, and a shift of 2 units downwards.
Question 6
- The parabola f intersects the x-axis at B and C, and the y-axis at E. The axis of symmetry of the parabola is x = 3.
- Determine the coordinates of point C by substituting the x-coordinate of C (x = 7) into the equation g(x) = -x/2 + 7/2.
- Calculate the x-coordinate of point B using the fact that the axis of symmetry is x = 3 and the distance between B and C is the same as the distance between C and the axis of symmetry.
- The equation of the parabola, f, can be determined in the form y = a(x-p)² + q, where p and q are the coordinates of the vertex of the parabola.
- Determine the equation of the graph of h, which is the reflection of f in the x-axis, simply by changing the sign of the y-values of f.
- Determine the maximum value of t(x) = 1-f(x) by understanding that this represents a reflection of f in the x-axis followed by a vertical shift upwards by 1 unit. Look at the maximum value of f.
- Solve the equation f(x² - 2) = 0 by substituting x² - 2 for x in the equation of f and then solving the resulting quadratic equation.
Question 7
- The function f(x) = (1/3)^x is a decreasing function because the base (1/3) is less than one.
- Determine the inverse of f(x). Substitute y for f(x) and rearrange the equation to solve for x in terms of y. Then swap x and y to obtain the inverse function.
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Description
This quiz delves into the analysis of functions, focusing on vertical and horizontal asymptotes, intercepts, and transformations of graphs. You will also explore the intersections of parabolas with axes and the properties of quadratic functions. Test your understanding of these critical algebraic concepts.
