This is the equation of a parabola. As you've written, the x intercepts are (1, 0) and
(5, 0).
Every parabola has an axis of symmetry which is the line that runs down its 'center'. This line divides the graph into two perfect halves.
There are two different formulas that you can use to find the axis of symmetry. One formula works when the parabola's equation is in vertex form and the other works when the parabola's equation is in standard form.
• If your equation is in vertex form, then the axis of is:
. x= h in the general vertex form equation y = (x-h)2 + k
• If your equation is in standard form, then the formula for the axis of symmetry is:
. x = -b/2a from the general standard form equation y = ax2+bx + c
This parabola's equation is in standard form; then the axis of symmetry is the line x =3.
The vertex of a parabola is the highest or lowest point, also known as the maximum or minimum of a parabola. It is the turning point of the parabola. The axis of symmetry intersects the vertex
How to find the vertex? It depends on whether the equation is in vertex or standard form .
• If your equation is in standard form, the x-coordinate of the vertex can be found by
. the formula:
. -b/2a. To get the y value of the vertex, just substitute -b/2a, into the equation.
• If your equation is in vertex form, the vertex is just (h,k) from the equation.
. (It's called 'vertex form' for a reason!)
• If 'a' is positive then the parabola opens upwards like a regular "U".
• If 'a' is negative, then the graph opens downwards like an upside down "U".
You probably know that the smaller |a| in the standard form equation of a parabola, the wider
the parabola. In other words y = .1x² is a wider parabola than y = .2x² and y = -.1x² is a wider parabola than y = .-2x² . You can understand this 'widening' effect in terms of the focus and directrix. As the distance between the focus and directrix increases, |a| decreases which means the parabola widens.
• If |a| < 1, the graph of the parabola widens. This just means that the "U" shape of the
. parabola stretches out sideways .
• If |a| > 1, the graph of the parabola becomes narrower (The effect is the opposite of
. |a| < 1).
A parabola is a locus of points equidistant from a single point, called the focus of the parabola, and a line, called the directrix of the parabola. This means that any point on the parabola must be just as far away from the directrix and from the focus .
Then:
Vertex: (3,4)
Focus: (3,15/4)
Directrix: y=17/4
Axis of Symmetry: x=3
In order to find the general shape of a parabola and be able to graph it, try plotting points by substituting in values for one variable, usually x, then solving for the other variable (the y value).
In this case, for y = -x² + 6x - 5:
• x = 0 ---→ y = -5
• x = 1 ---→ y = 0
• x = 2 ---→ y = 3
• x = 3 ---→ y= 4
• x = 4 ---→ y= 3
• x = 5 ---→ y= 0
• x = 6 ---→ y= -5
• x = 7 ---→ y= -12
• x = -1 --→ y= -12
• x = -2 --→ y= -21
Hope, using all this information, try to graph this nice parabola. If you find yourself in trouble, you can see the graph here:
Answers & Comments
Verified answer
Hi, Hope
This is the equation of a parabola. As you've written, the x intercepts are (1, 0) and
(5, 0).
Every parabola has an axis of symmetry which is the line that runs down its 'center'. This line divides the graph into two perfect halves.
There are two different formulas that you can use to find the axis of symmetry. One formula works when the parabola's equation is in vertex form and the other works when the parabola's equation is in standard form.
• If your equation is in vertex form, then the axis of is:
. x= h in the general vertex form equation y = (x-h)2 + k
• If your equation is in standard form, then the formula for the axis of symmetry is:
. x = -b/2a from the general standard form equation y = ax2+bx + c
This parabola's equation is in standard form; then the axis of symmetry is the line x =3.
The vertex of a parabola is the highest or lowest point, also known as the maximum or minimum of a parabola. It is the turning point of the parabola. The axis of symmetry intersects the vertex
How to find the vertex? It depends on whether the equation is in vertex or standard form .
• If your equation is in standard form, the x-coordinate of the vertex can be found by
. the formula:
. -b/2a. To get the y value of the vertex, just substitute -b/2a, into the equation.
• If your equation is in vertex form, the vertex is just (h,k) from the equation.
. (It's called 'vertex form' for a reason!)
• If 'a' is positive then the parabola opens upwards like a regular "U".
• If 'a' is negative, then the graph opens downwards like an upside down "U".
You probably know that the smaller |a| in the standard form equation of a parabola, the wider
the parabola. In other words y = .1x² is a wider parabola than y = .2x² and y = -.1x² is a wider parabola than y = .-2x² . You can understand this 'widening' effect in terms of the focus and directrix. As the distance between the focus and directrix increases, |a| decreases which means the parabola widens.
• If |a| < 1, the graph of the parabola widens. This just means that the "U" shape of the
. parabola stretches out sideways .
• If |a| > 1, the graph of the parabola becomes narrower (The effect is the opposite of
. |a| < 1).
A parabola is a locus of points equidistant from a single point, called the focus of the parabola, and a line, called the directrix of the parabola. This means that any point on the parabola must be just as far away from the directrix and from the focus .
Then:
Vertex: (3,4)
Focus: (3,15/4)
Directrix: y=17/4
Axis of Symmetry: x=3
In order to find the general shape of a parabola and be able to graph it, try plotting points by substituting in values for one variable, usually x, then solving for the other variable (the y value).
In this case, for y = -x² + 6x - 5:
• x = 0 ---→ y = -5
• x = 1 ---→ y = 0
• x = 2 ---→ y = 3
• x = 3 ---→ y= 4
• x = 4 ---→ y= 3
• x = 5 ---→ y= 0
• x = 6 ---→ y= -5
• x = 7 ---→ y= -12
• x = -1 --→ y= -12
• x = -2 --→ y= -21
Hope, using all this information, try to graph this nice parabola. If you find yourself in trouble, you can see the graph here:
http://www.wolframalpha.com/input/?i=plot+y+%3D+%3...
Bye. Good luck!