Chapter 3 - Curve Sketching 1

Question 1

How do you find the roots of a graph in the form y = (ax+b)/(cx+d)? e.g, where the curve crosses the x axis?

Answer 1

To find the roots of a graph in the form y = (ax+b)/(cx+d), you set y to 0, and calculate the values for x.

Question 2

How do you find the y-intercepts of a graph in the form y = (ax+b)/(cx+d)? e.g, where the curve crosses the y axis?

Answer 2

To find the y-intercepts of a graph in the form y=(ax+b)/(cx+d), you set x to 0, and calculate the values for y.

Question 3

How do you find the vertical asymptote of a graph in the form y = (ax+b)/(cx+d)?

Answer 3

To find the vertical asymptote of a graph in the form y=(ax+b)/(cx+d), you set the denominator to 0.

Question 4

How do you find the horizontal asymptote of a graph in the form y = (ax+b)/(cx+d)?

Answer 4

To find the horizontal asymptote of a graph in the form y=(ax+b)/(cx+d), you consider large values x, for which y tends towards a value.

Question 5

How do you find the roots of a graph in the form y = (ax² + bx + c)/(dx²+ex + f)? e.g, where the curve crosses the x axis?

Answer 5

To find the roots of a graph in the form y = (ax² + bx + c)/(dx²+ex + f), you set y to 0, and calculate the values for x.

Question 6

How do you find the y-intercepts of a graph in the form y = (ax² + bx + c)/(dx²+ex + f)? e.g, where the curve crosses the y axis?

Answer 6

To find the y-intercepts of a graph in the form y = (ax²+ bx + c)/(dx²+ex + f), you set x to 0, and calculate the values for y.

Question 7

How do you find the horizontal asymptote of a graph in the form y = (ax² + bx + c)/(dx²+ex + f)?

Answer 7

To find the horizontal asymptote of a graph in the form y = (ax² + bx + c)/(dx²+ex + f), you consider large values x, for which y tends towards a value.

Question 8

How do you find the vertical asymptote of a graph in the form y = (ax² + bx + c)/(dx²+ex + f)?

Answer 8

To find the vertical asymptote of a graph in the form y = (ax² + bx + c)/(dx²+ex + f), you set the denominator to 0.

Question 9

Define Polar Coordinates.

Answer 9

A point on a region, P, can be defined as (x,y), in Cartesian format.

The polar coordinates of a point, P, are defined as the distance from the origin, r, and the counterclockwise angle from the positive real axis, theta. (r, θ).

The conditions are:

r ≥ 0, and 0 ≤ θ < 2π

Question 10

How do you convert Polar to Cartesian coordinates, and vice versa?

Answer 10

Cartesian Coordinates are (x,y), and Polar Coordinates are (r,θ)

To convert from Polar to Cartesian:

x = r cos(θ)
y = r sin(θ)

To convert from Cartesian to Polar:

r² = x² + y²
tan(θ) = y/x

Question 11

What is the equation of a parabola?

Answer 11

The equation y = k(x-h)² + c describes a parabola with its vertex at (h,c)

The equation x = k(y-h)² + c describes a parabola with its vertex at (c,h)

Question 12

What is the equation of an ellipse?

Answer 12

The equation of an ellipse, centred at the origin is:

(x-h)²/(a²) + (y-k)²/(b²) = 1, where the ellipse will be centred on the point (h,k), with radius of a in the x-direction and b in the y-direction.

Question 13

What is the equation of a hyperbola?

Answer 13

The equation of the form (x-h)²/(a²) - (y-k)²/(b²) = 1 describes a hyperbola centred on (h,k) with asymptotes y = (±b/a)(x-h)+k, roots (h±a,0), and y intercepts are found when x is set to 0.

Question 14

What equation describes a rectangular hyperbola?

Answer 14

An equation of the form (x-h)(y-k) = c² describes a rectangular hyperbola centred on (h,k) with asymptotes x = h, and y = k.

Question 15

Define sinh x, cosh x and tanh x as exponentials.

Answer 15

sinh x = (eˣ - e⁻ˣ)/2

cosh x = (eˣ + e⁻ˣ)/2

tanh x = sinh x / cosh x = (eˣ - e⁻ˣ)/(eˣ + e⁻ˣ)

Question 16

Name the hyperbolic identity.

Answer 16

The hyperbolic identity: cosh² x - sinh² x = 1

Question 17

What are the inverse hyperbolic functions?

Answer 17

sinh⁻¹ x = ln(x + sqrt(x² + 1)) for all x

cosh⁻¹ x = ln(x + sqrt(x² - 1)); x≥1

tanh⁻¹ x = 1/2 ln((1+x)/(1-x)); -1 < x < 1