To Find:
The vertex , axis of symmetry and y − intercept.
Answer to Problem 9CYU
Vertex : (1,2)
Axis of Symmetry : x = 1
y-intercept : -1
Explanation of Solution
Given Information:
Given equation is :
y=−3x2+6x−1
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−1 ; comparing it with standard form of quadratic equation , we have a = -3 , b = 6 , c = -1 .
- Find 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 :
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 =1 in the equation y=−3x2+6x−1
y=−3x2+6x−1y=−3(1)2+6(1)−1.....................[x=1]y=−3+6−1................................[simplify]y=2
The vertex is at (1,2)
- Find y-intercept :
The y-coordinate of the point at which the graph cuts the y-axis is the y-intercept.
Hence , substituting x=0 in the equation y=−3x2+6x−1 , we will get the y-intercept.
y=−3x2+6x−1y=−3(0)2+6(0)−1..........................[x=0]y=−1
Or
y-intercept always occurs at (0,c) and c = -1 here.
So, y-intercept = -1 and is located at (0,-1)
Chapter 9 Solutions
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
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