
Concept explainers
To Graph:
The given function.

Explanation of Solution
Given Information:
Given equation is :
y=−3x2+6x−4
Concept Used :
- Quadratic Functions:
- Quadratic functions are non-linear and are of the form:
f(x)=ax2+bx+c;a≠0
This is the standard form of Quadratic function.
- Shape of the graph of a Quadratic function is called Parabola.
- Parabolas are symmetric about a central line called Axis Of Symmetry.
- The axis of symmetry intersects a parabola only at one point , called the Vertex.
- Parent Function:
y=x2
- Standard Form :
f(x)=ax2+bx+c;a≠0
- Type of Graph:
Parabola
- Axis of Symmetry: Passes through the vertex and divides the parabola into two congruent halves.
x=−b2a
- y-intercept = The y coordinate of the point at which the graph intersects y-axis = c
- Graph:
- If a > 0 ,
- The graph of ax2+bx+c opens upward.
- The lowest point on the graph is the minimum.
- If a < 0 ,
- The graph of ax2+bx+c opens downward.
- The highest point on the graph is the maximum.
- The maximum or the minimum is the vertex.
Calculation:
- In the equation y=−3x2+6x−4 ; comparing it with standard form of quadratic equation , we have a = -3 , b = 6 , c = -4 .
- Find equation of Axis of Symmetry :
y=ax2+bx+c...........................[standard_form]x=−b2a.....................................[formula_of_axis_of_symmetry]x=−62(−3)=1..........................[a=−3,b=6]
The equation of axis of symmetry is x=1
- Find the Vertex and determine whether it is maximum or minimum:
Since axis of symmetry passes through the vertex , we have x coordinate of the vertex (x,y) is x=1
Substituting the value of x=1in the equation y=−3x2+6x−4
y=−3x2+6x−4y=−3(1)2+6(1)−4...............[x=1]y=−3+6−4..................................[simplify]y=−1
The vertex is at (1,−1) .
Since the a = -3 < 0 .So, the graph opens downward .Hence, the vertex is the maximum.
- Find y-intercept :
The y-coordinate of the point at which the graph cuts the y-axis is the y-intercept.
y-intercept always occurs at (0,c) and c = -4 here.
So, y-intercept = -4 and is located at (0,-4).
- The axis of symmetry cuts the parabola into two equal parts . So, if there is a point on one side, there is a corresponding point on the other side that is at a same distance from the axis of symmetry and the same y-value.
x | y |
-1 | -13 |
0 | -4 |
1 | -1 |
2 | -4 |
3 | -13 |
- Connect the points with a smooth curve.
Graph :
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