Module 6: Quadratic Graphs - Equations, Solutions, and Graphs

This assignment delves into the fundamentals of quadratic graphs, starting with the basic form of a quadratic equation, identifying constants and variables, and sketching parabolas based on the sign of 'a'. It explores the role of the 'c' constant in determining the vertical position of the parabola. The assignment then moves on to solving quadratic equations using the quadratic formula, emphasizing the importance of setting the equation to zero and finding intercepts. It provides a practical example using the equation y = 2x^2 + 4x - 3 and guides the student to calculate the discriminant to determine the nature of the roots (real or complex) based on its value. Finally, the assignment explains the concept of the axis of symmetry and vertex coordinates for a quadratic equation and asks the student to find these values and graph the equation.

1) Graph Basics
a)
The general form of quadratic equation is given by,
y ( x )=a x2+bx+ c
Where a,b,c are constants and x and y are variables.
b)
The line of symmetry of this parabola is y axis.
c)

The vertex of this parabola is (0,0)
d)
the constant c determines how vertical distance the parabola moves.
That is the distance that the vertex moves with respect to the y axis of the graph.
2)
a) Befrore solving a quadratic equation, you must make it equal to zero.
b)To solve a quadratic equation usinga graph find the intercepts.
c)
given,
y= 2 x2 + 4 x−3
here a=2, b=4, c=-3
d)
=2 x2 + 4 x−3=0
= x=−b ± √ b2−4 ac
2 a
= x=−4 ± √ 16+24
4

=x=−1+ √ 5
2 , -1-
√ 5
2
That is x=0.58,-2.58
e)
discriminant, D= √ b2−4 ac
=√40
i)
if the discriminant is positive, there will be 2 values for x and there fore two separate roots
ii)
if the discriminant is zero, there is only one value for x, but there are in fact two roots which are
equal.
iii)
if the discriminant is negative, there is no real value for x. The value for the roots is a complex
number.
3.
a)
For a quadratic equation in standard for, y=a x2 +bx+ c.
the axis of symmetry is a vertical line x=−b
2 a
b)the y cordinate of the vertex is found out by substituting the above value of x the general
quadratic equation.
That is,
= y= a∗b2
4 a2 − b2
2 a +c
= y=−b2
4 a +c
Therefore the cordinates = (−b
2 a , −b2
4 a +c ¿
c)
i)does the graph turn up or down
ii)find the vertex
iii)find the intercepts
iv)graph the parabola