Find equation of a quadratic with line of symmetry

In summary, to find the equation of a quadratic with a given line of symmetry, you must first identify the coordinates of the vertex and then use the formula y = a(x - h)^2 + k. The line of symmetry in a quadratic equation is significant as it divides the parabola into two equal halves and is perpendicular to the axis of symmetry. A quadratic equation can only have one line of symmetry, which is always at the midpoint between the two roots and is not affected by the value of 'a'. The line of symmetry can be found without graphing the equation by using the quadratic formula or factoring the equation, or by finding the coordinates of the vertex using the formula h = -b/2a.
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Here's something to start you off. The axis of symmetry for the parabola \(\displaystyle y = ax^2 + bx + c\) is \(\displaystyle x = -\dfrac{b}{2a}\) and if an x-intercept is at (-2, 0) then \(\displaystyle 0 = a(-2)^2 + b(-2) + c\). That will give you both b and c as functions of a.

Can you finish?

-Dan
 
  • #3
Equivalently, if the axis of symmetry is x= 1.75 then the equation can be written as $y= a(x- 1.75)^2+ c= ax^2+ 3.5ax+ 3.0625+ c= ax^2+ bx+ 22. We must have b= 3.5a.

If, in addition, y= 0 when x= -2, 0= 4a- 2b+ 22 so b= 11+ 2a.
 

FAQ: Find equation of a quadratic with line of symmetry

What is a line of symmetry in a quadratic equation?

A line of symmetry in a quadratic equation is a vertical line that divides the parabola into two equal halves. This line passes through the vertex of the parabola and is perpendicular to the x-axis.

How do I find the line of symmetry of a quadratic equation?

The line of symmetry of a quadratic equation can be found by using the formula x = -b/2a, where a and b are the coefficients of the quadratic equation in standard form (ax^2 + bx + c).

What is the significance of the line of symmetry in a quadratic equation?

The line of symmetry is significant because it helps us determine the vertex of the parabola, which is the highest or lowest point on the curve. It also divides the parabola into two equal halves, making it easier to graph and solve the equation.

Can the line of symmetry be a horizontal line?

No, the line of symmetry for a quadratic equation is always a vertical line. This is because the parabola is symmetric about the y-axis, not the x-axis.

How do I find the equation of a quadratic with a given line of symmetry?

To find the equation of a quadratic with a given line of symmetry, you can use the formula y = a(x-h)^2 + k, where (h,k) is the coordinates of the vertex and a is the coefficient of the squared term. Plug in the coordinates of the vertex and the given line of symmetry to solve for a, and then substitute the value of a into the formula to get the final equation.

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