√ How To Arrange A Function Quadratics Of The Graph

At this time I would give a tutorial and the way how to make a graph of graphic function. This tutorial i ensure will give the best solution for student learning mathematics.

Okey lets we do.

This is the formulas needed to make equation quadratic function.

Case 1

If the graph sketch through the vertex /extreme (p , q) and any point (x , y). The graph like this

At this time I would give a tutorial and the way how to make a graph of graphic function √ How to arrange a function quadratics of the graph we get the formulas : $At this time I would give a tutorial and the way how to make a graph of graphic function √ How to arrange a function quadratics of the graph$

Case 2

When the graph of function across X axis we say $\left( { x }_{ 1 },0 \right) \quad and\quad \left( { x }_{ 2 },0 \right) \\$ , and through any point $\left( { x },y \right) \\$ then we use this formula :

At this time I would give a tutorial and the way how to make a graph of graphic function √ How to arrange a function quadratics of the graph we get the formula : $\boxed { y=a\left( x-{ x }_{ 1 } \right) \left( x-{ x }_{ 2 } \right) } \\$

Lets we see the examples below.

Example 1

Identify the graph function of quadratic equation of the picture below.

At this time I would give a tutorial and the way how to make a graph of graphic function √ How to arrange a function quadratics of the graph

Alternative Answer

We can see that the graph through the vertex and sure by using the formula case 1

$Vertex:\left( -1,2 \right) \quad and\quad any\quad point\quad \left( 0,4 \right) \\$

$y-q=a{ \left( x-p \right) }^{ 2 }\\ \\ 4-2=a{ \left( 0-\left( -1 \right) \right) }^{ 2 }\\ \\ 2=a.1\quad \Longrightarrow \quad a=2$

Substitute (a) to the origin formula then we get

$y-q=a{ \left( x-p \right) }^{ 2 }\\ \\ y-2=2\left( x-\left( -1 \right) \right) ^{ 2 }\\ \\ y-2=2{ \left( x+1 \right) }^{ 2 }\\ \\ y-2=2\left( x+1 \right) \left( x+1 \right) \\ \\ y-2=2\left( { x }^{ 2 }+2x+1 \right) \\ \\ y-2=2x^{ 2 }+4x+2\\ \\ y=2x^{ 2 }+4x+4$

so the equation of the graph is $\\ y=2x^{ 2 }+4x+4$

Example 2

Identify the graph function of quadratic equation of the picture below.

At this time I would give a tutorial and the way how to make a graph of graphic function √ How to arrange a function quadratics of the graph

Alternative answer

We can see that the graph across x axis and through any point and sure by using the formula case 2

$\\ intersection\quad of\quad x\quad axis\quad :\quad \left( -4,0 \right) \quad and\quad \left( 3,0 \right) \quad \\ any\quad point\quad :\quad \left( 0,4 \right)$

$\\ y=a\left( x-{ x }_{ 1 } \right) \left( x-{ x }_{ 2 } \right) \\ \\ 4=a\left( 0-(-4) \right) \left( 0-3 \right) \\ \\ 4=a\left( 4 \right) \left( -3 \right) \\ 4=-12a\quad \Longrightarrow \quad a=-\frac { 1 }{ 3 } \\$

Substitute (a) to the origin formula then we get

$\\ y=a\left( x-{ x }_{ 1 } \right) \left( x-{ x }_{ 2 } \right) \\ \\ y=-\frac { 1 }{ 3 } \left( x-(-4) \right) \left( x-3 \right) \\ \\ y=-\frac { 1 }{ 3 } \left( x+4 \right) \left( x-3 \right) \\ \\ y=-\frac { 1 }{ 3 } \left( { x }^{ 2 }+x-12 \right) \\ \\ y=-\frac { 1 }{ 3 } { x }^{ 2 }-\frac { 1 }{ 3 } x+4\\ so\quad the\quad equation\quad of\quad the\quad graph\quad is\quad y=-\frac { 1 }{ 3 } { x }^{ 2 }-\frac { 1 }{ 3 } x+4\\$

Example 3

Identify the graph function of quadratic equation of the picture below.

At this time I would give a tutorial and the way how to make a graph of graphic function √ How to arrange a function quadratics of the graph

Alternative answer

We can see that the graph through the vertex and any point x axis and sure by using the formula case 1

$\\ vertex\quad :\quad \left( 3,8 \right) \quad any\quad point\quad (5,0)\\ \\ y-q=a{ \left( x-p \right) }^{ 2 }\\ \\ 0-8=a\left( 5-3 \right) ^{ 2 }\\ \\ -8=a\left( 2 \right) ^{ 2 }\\ \\ -8=4a\quad \Longrightarrow \quad a=-2\\$

Substitute (a) to the origin formula then we get

$\\ y-q=a{ \left( x-p \right) }^{ 2 }\\ \\ y-8=-2\left( x-3 \right) ^{ 2 }\\ \\ y-8=-2\left( x-3 \right) \left( x-3 \right) \\ \\ y-8=-2\left( { x }^{ 2 }-6x+9 \right) \\ \\ y-8=-2{ x }^{ 2 }+12x-18\\ \\ y=-2{ x }^{ 2 }+12x-10\\ \\ so\quad the\quad equation\quad of\quad the\quad graph\quad is\quad y=-2{ x }^{ 2 }+12x-10\\$

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