Y9 Straight Line Activity (draft)

See the link below for an activity designed to explore the equation y = mx + c  

https://student.desmos.com/join/afvs6t

Circle Theorems (Draft)

Circle Theorems

Test

See this applet, CT2a, on Geogebra website here

The Parabola

A parabola is a U-shaped curve. The intersection of a cone and a plane parallel to the slant height of the cone is parabolic. A quadratic relation forms a parabola, but it can also be defined independently of this. 

A Focus and a horizontal Directrix

A parabola is the set of all points that are the same distance from a given point \(F(a, b)\) and a given line \(y = c\). The given point is called the \(Focus\), and the line the \(Directrix\).

An interesting property you can explore using the link above is that light from an infinite source will always be reflected to the Focus.

A Focus and any Directrix

A parabola is the set of all points that are the same distance from a given point \(F(a, b)\) and a given line \(y = cx+d\). The given point is called the Focus, and the line the Directrix.

Hyperbolic Graph

 A hyperbola can be defined in multiple ways

1) Using two points

A hyperbola is a set of points, such that for any point \(P\) of the set, the absolute difference of the distances from two points, \(F_1\) and \(F_2\), is constant \[||PF_1|-|PF_2|| = 2a\] \(a\) is the distance from the midpoint to the vertices, \(V_1\) and \(V_2\). The hyperbola has asymptotes at \[y = ± \frac{b}{a}x\]

Hyperbolic Curve. Drag the shaded blue circles to adjust the parameters.

2) Using a circle and a point

If \(c_2\) is the circle with midpoint \(F_2\) and radius \(2a\),then the distance of a point \(P\) of the right branch of the hyperbola to the circle \(c_2\) equals the distance to the focus \(F_1\). If \(N\) is the point on \(c_2\) this can be written:\[|NP| = |PF_1|\] \(c_2\)is called the circular directrix (related to focus \(F_2\)) of the hyperbola

Hyperbolic Curve. Drag the shaded green circle to adjust the parameters.

Quadratic Graph

The general form of a quadratic is \[y = ax^2 + bx + c\]

Using the link below, you can explore some key features of quadratic graphs including:

• Roots, the y-intercept and the line of symmetry
• The family of quadratics of the form \(y = ax^2\) 
• The family of quadratics of the form \(y = x^2 + bx\)
• The family of quadratics of the form \(y = x^2 + c\)
• The completed square form \(y = a(x+p)^2 + q\) (as shown)

 

Straight Line Graph

A useful form of the equation of a straight line graph is \[y=mx + c\] where \(m\) is the gradient and \(c\) is the y-intercept. 

Using the link below you can explore:

• The effect of changing \(m\)
• The effect of changing \(c\)
• The family of graphs of the form \(y = mx\)
• The family of graphs of the form \(y = x + c\)