Graphical Representation of Quadratic Equation

A quadratic graph depicts a U-shaped curve drawn for a quadratic function. In Mathematics, a parabola is one of the conic sections, which is formed by the intersection of a right circular cone by a plane surface. It is a symmetrical plane U-shaped curve. A parabola graph whose equation is in the form of f(x)=a x^2+b x+c is the standard form of a parabola. The vertex of a parabola is the extreme point in it whereas the vertical line passing through the vertex is the axis of symmetry.

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In this article, we will cover the concept of the graph of quadratic equations. This concept falls under the broader category of complex numbers and quadratic equations, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

What is a Parabola Graph?

A parabola is a U-shaped curve that is drawn for a quadratic function, f(x) = . The graph of the parabola is downward (or opens down) when the value of a is less than 0, a < 0. The graph of parabola is upward (or opens up) when the value of

$a$ is more than $0, a>0$. Hence, the direction of a parabola is determined by the sign of coefficient ' $a$ '.

How to graph a quadratic function
We have $y=f(x)=a x^2+b x+c$ where $a, b, c \in R$ and $a \neq d$
Expression $y=a x^2+b x+c=f(x)$ can be represented as

$$
y=a\left(x^2+\frac{b}{a} x+\frac{c}{a}\right)
$$

which on further simplification is converted in the form of

$$
\begin{aligned}
& \Rightarrow y=a\left[\left(x+\frac{b}{2 a}\right)^2-\frac{D}{4 a^2}\right] \\
& \text { or }\left(y+\frac{D}{4 a}\right)=a\left(x+\frac{b}{2 a}\right)^2
\end{aligned}
$$

Now, let $y+\frac{D}{4 a}=Y$ and $x+\frac{b}{2 a}=X$

$$
\therefore \mathrm{Y}=\mathrm{aX}^2
$$

The shape of the $y=f(x)$ will be parabolic
Vertex of the parabola will be $\left(\frac{-\mathrm{b}}{2 \mathrm{a}}, \frac{-\mathrm{D}}{4 \mathrm{a}}\right)$

$$
\left[y+\frac{D}{4 a}=0 \Rightarrow y=-\frac{D}{4 a}\right]
$$

If the parabola opens upward (when a > 0) then the y value of the vertex represents the least value of the equation, and if opens downward (when a < 0) then the y value of the vertex represents the greatest value of the parabola.

Both least and greatest values are attained at the x value of the vertex of the parabola

Hence the graph of any general quadratic equation will look like the below graph (given a>0)

In the general quadratic equation if $y=a x^2+b x+c=f(x)$ and if $a>0$
Then the parabola opens upward. As given below,

if a < 0 it opens downward. As given below,

Summary

Quadratic graphs are an important topic in mathematics, characterized by their parabolic shape and specific properties. Mastering the vertex, axis of symmetry, intercepts, and discriminant helps in graphing and interpreting quadratic functions effectively. These graphs are not only essential in mathematical theory but also have extensive applications in science, engineering, economics, and various fields requiring optimization and modeling of curved trajectories. The face of the graph determine its opening that can be closedly understood by the sign of + or - if it is on right hand side or left hand side. They are also used for the design in architecture like bridges and other infrastructure along with diverse application in optics and visual devices. They are designed in such a way so that the structures unable to stand the maximum amount of stress which can be distributed evenly on the surface to avoid any hazardous condition.

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Solved Examples Based on Graph of Quadratic Equations:

Example 1: $f(x)=2 x^2+a x+2$ if $f(x)=0$ has no real root then ' $a^{\prime}$ takes value lying in an interval
1) $(-5,5)$
2) $(-4,4)$
3) $(-6,6)$
4) $(-7,7)$

Solution:
As we learned in
Quadratic Expression Graph when $\mathrm{a}>0 \& \mathrm{D}<0$ -
No Real and Equal root of

$$
\begin{aligned}
& f(x)=a x^2+b x+c \\
& \& D=b^2-4 a c \\
& \text { - wherein }
\end{aligned}
$$

$\because f(x)=0$ has no real root, so $D<0$

$$
\therefore a^2-16<0 \Rightarrow a \epsilon(-4,4)
$$

$\therefore$ Option B
Example 2: If equations $a x^2+b x+c=0,(a, b, c \in R, a \neq 0)$ and $2 x^2+3 x+4=0$ Have a common root, then a:b:c equals :
1) $1: 2: 3$
2) $2: 3: 4$
3) $4: 3: 2$
4) $3: 2: 1$

Solution:
As we have learned
Quadratic Expression Graph when a $>0$ \& $<0$ -
No Real and Equal root of

$$
\begin{aligned}
& f(x)=a x^2+b x+c \\
& \& D=b^2-4 a c
\end{aligned}
$$

- wherein

Condition for both roots common -

$$
\begin{aligned}
& \frac{a}{a^{\prime}}=\frac{b}{b^{\prime}}=\frac{c}{c^{\prime}} \\
& \text { - wherein } \\
& a x^2+b x+c=0_{\&} \\
& a^{\prime} x^2+b^{\prime} x+c^{\prime}=0
\end{aligned}
$$

are the 2 equations

$$
2 x^2+3 x+4=0 \text { has determinant }=9-32=-23<0
$$

So, non-real roots which means both roots are common (as complex roots occur in conjugate )
So, a:b:c $\$=2: 3: 4 \$$
Example 3: The value of $\$$ llambda\$ such that the sum of the squares of the roots of the quadratic equation, $\$ x^{\wedge} 2 \$$
Solution:
Sum of Roots in Quadratic Equation -

$$
\begin{aligned}
& a x^2+b x+c=0 \\
& a, b, c \in C
\end{aligned}
$$

Product of Roots in Quadratic Equation -

$$
\alpha \beta=\frac{c}{a}
$$

- wherein
$\alpha$ and $\beta$ are roots of a quadratic equation:

$$
\begin{aligned}
& a x^2+b x+c=0 \\
& a, b, c \in C
\end{aligned}
$$

Quadratic Expression Graph when $a>0 \& \mathrm{D}<0$ -
No Real and Equal root of

$$
\begin{aligned}
& f(x)=a x^2+b x+c \\
& \& D=b^2-4 a d \\
& \text { - wherein }
\end{aligned}
$$

Given quadratic equation

$$
x^2+(3-\lambda) x+2=\lambda
$$

roots are $\alpha$ and $\beta$
from the concept

$$
\begin{aligned}
\alpha+\beta & =\lambda-3 \text { and } \alpha \beta=2-\lambda \\
\alpha^2+\beta^2 & =(\alpha+\beta)^2-2 \alpha \beta \\
& =\lambda^2+9-6 \lambda-4+2 \lambda \\
& =\lambda^2-4 \lambda+5 \\
& =(\lambda-2)^2+1
\end{aligned}
$$

least value when $\lambda=2$
Hence, the answer is 2 .
Example 4: Let $f(x)=x^2+2(a-1) x+(a+5)$, then the values of ' $a^{\prime}$ for which $f(x)=0$ doesn't have two real and distinct roots is
1) $(-1,4]$
2) $(-1,4)$
3) $(-1,4)$
4) $[-1,4]$

Solution:

Solution:
As we learned in
Quadratic Expression Graph when $\mathrm{a}>0$ \& $\mathrm{D}=0$ -
Real and Equal roots of

$$
f(x)=a x^2+b x+c
$$

\& $D=b^2-4 a c$
- wherein

$\because f(x)=0$ doesn't have real and distinct roots, so either it will have real and equal roots or imaginary roots.

$$
\begin{aligned}
& \text { So } D \leq 0 \Rightarrow 4\left(a^2-2 a+1\right)-4(a+5) \leq 0 \\
& \Rightarrow(a-4)(a+1) \leq 0 \Rightarrow a \epsilon[-1,4]
\end{aligned}
$$

Example 5: Let $f(x)=x^2+2(a-1) x+\left(a^2+1\right)$ then the values of ' $a$ ' for which $f(x)=0$ has real and equal roots is
Solution:
As we learned in
Quadratic Expression Graph when $a>0 \& D=0$ -
Real and Equal roots of

$$
\begin{aligned}
& f(x)=a x^2+b x+c \\
& \& D=b^2-4 a c
\end{aligned}
$$

- wherein

For $x^2+2(a-1) x+\left(a^2+1\right)=0$, to have real and equal roots, $D=0$

$$
\begin{aligned}
& \Rightarrow 4\left(a^2-2 a+1\right)-4\left(a^2+1\right)=0 \\
& \Rightarrow a=0
\end{aligned}
$$