Chapter 18 math notes, Study notes of Mathematics

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Maths Notes by Vasumitra

Gajbhiye

18. Curve graphs

18.1 Drawing quadratic graphs (the parabola)

Step 1 → make a table

Step 2 → plot a graph

Graph for positive coefficient of X form valley. Graph with valley have minimum point. Graph for negative coefficient of X form hill. Graph with hill have maximum point.

The axis of symmetry and the turning point

The axis of symmetry is the line which divides the parabola into two symmetrical halves. In the two graphs above, the y-axis (x = 0) is the axis of symmetry. The turning point or vertex of the graph is the point at which it changes direction or gradient = 0. For both of the graphs above, the turning point is at the origin (0, 0).

Equations in the form of

complete the table of values rule the axes and label them plot the (x, y) values join them with a smooth curve label the graph with its equation

Sketching quadratic functions

1. Identify the shape of the graph → Positive coefficient of X gives valley, negative coefficient of X gives hill.

y = x^2 + ax + b

In the above example, the coordinate of the turning point are (- -2, 7) = (2,7).

18.2 Drawing reciprocal graphs (the hyperbola)

Equations in the from of (where a is a whole number) are called reciprocal equations. The graphs of reciprocal equations are called hyperbolas. To plot a hyperbola:

1. Make a table
2. Plot the graph

y = a / x

An asymptote is a line that a graph approaches but never intersects. Draw asymptote using dotted line.

Sketching graphs of reciprocal functions

1. Identify the shape of the graph → if a>0, then graph forms in the first and third quadrant. if a<0 then the graph forms in the second and fourth quadrant.
2. Work out whether the graph intercepts the x-axis → if q≠0 , the graph will have one x-intercept. Make y = 0 in the equation to find the value of X-intercept.
3. Determine the asymptotes → The graph never intersects the y-axis (in IGCSE). The other is the line y = q. If q = 0, the x-axis is the other asymptote.
4. Using the asymptotes and the x-intercept, sketch and label the graph.

In summary, to solve a quadratic equation graphically:

read off the x co-ordinates of any points of intersection for the given y-values you may need to rearrange the original equation to do this.

18.4 Using graphs to solve simultaneous linear and

non-linear equations

18.5 Other non-linear graphs

Plotting cubic graphs

If the coefficient of the term is positive, the graph will take this shapes.

If the coefficient of the term is negative, the graph will

x^3

x^3

take this shapes.

Sketching cubic functions

1. Determine the orientation of the graph → use the technique given above.
2. Find the y-intercept → substitute x =0 in the equation.
3. Find the x-intercept → When the cubic equation is given in factor form (for example y = (x + a) (x + b)(x + c), you can let y = 0 and solve for x.
4. Find the turning point of the graph → you’ll learn this later when using differentiation.

Exponential graphs

They have a general formula of. The graph always intersect the y-axis at the point (0,1) because for all values of a.

Recognising which graph to draw

y = ax

a^0 = 1

Special case

Equations of Tangents

1. Find the gradient → use dy/dx to find the gradient. Substitute the value of x given in the question
2. If y coordinate is not given then find the y-coordinate by substituting the value of x in the equation.
3. Now you have the gradient(m) the x and y coordinate, substitute all three into the standard form equation and find the value of c.
4. Now you have the equation of tangent, write the final answer.

Turning Points

1. Find the dy/dx of the given equation.
2. Find the value of x where dy/dx = 0. this value of x is the x-coordinate of the turning point.
3. Find the y-coordinate of the turning point, by substituting the value of x-coordinate of turning point into the equation.

y = mx + c

4. Now you have the coordinates of the turning point. Remember when dy/dx = 0 it is the turning point.

Maximum and Minimum Points