Quadratic equations form a parabola when graphed. The parabola can open upward or downward, and it can shift up or down or horizontally, depending on the constants of the equation when you write it in the form y = ax squared + bx + c. The variables y and x are graphed on the y and x axes, and a, b and c are constants. Depending on how high the parabola is located on the y-axis, an equation may have zero, one or two x-intercepts but it will always have one y-intercept.
Graph several parabolas changing only one of the three constants to see what affect each one has on the position and shape of the parabola.
If you mix up the x and y axes or the x and y variables, the parabolas will be horizontal instead of vertical.
Check to make sure your equation is a quadratic equation by writing it in the form y = ax squared + bx + c where a, b, and c are constants and a is not equal to zero. Find the y-intercept for the equation by letting x equal zero. The equation becomes y = 0x squared + 0x + c or y = c. Note that the y-intercept of a quadratic equation written in the form y = ax squared + bx = c will always be the constant c.
To find the x-intercepts of a quadratic equation, let y = 0. Write down the new equation ax squared + bx + c = 0 and the quadratic formula that gives the solution as x = -b plus or minus the square root of (b squared - 4ac), all divided by 2a. The quadratic formula can give zero, one or two solutions.
Solve the equation 2x squared - 8x + 7 = 0 to find two x-intercepts. Place the constants into the quadratic formula to get -(-8) plus or minus the square root of (-8 squared - 4 times 2 times 7), all divided by 2 times 2. Calculate the values to get 8 +/- square root (64 - 56), all divided by 4. Simplify the calculation to get (8 +/- 2.8)/4. Calculate the answer as 2.7 or 1.3. Note that this represents the parabola crossing the x-axis at x = 1.3 as it decreases to a minimum and then crosses again at x = 2.7 as it increases.
Examine the quadratic formula and note that there are two solutions because of the term under the square root. Solve the equation x squared + 2x +1 = 0 to find the x-intercepts. Calculate the term under the square root of the quadratic formula, the square root of 2 squared - 4 times 1 times 1, to get zero. Calculate the rest of the quadratic formula to get -2/2 = -1, and note that if the term under the square root of the quadratic formula is zero, the quadratic equation has only one x-intercept, where the parabola just touches the x-axis.
From the quadratic formula, note that if the term under the square root is negative, the formula has no solution and the corresponding quadratic equation will have no x-intercepts. Increase c, in the equation from the previous example, to 2. Solve the equation 2x squared + x + 2 = 0 to get the x-intercepts. Use the quadratic formula to get -2 +/- square root of (2 squared - 4 times 1 times 2), all divided by 2 times 1. Simplify to get -2 +/- square root of (-4), all divided by 2. Note the square root of -4 has no real solution and so the quadratic formula shows that there are no x-intercepts. Graph the parabola to see that increasing c has raised the parabola above the x-axis so that the parabola no longer touches or intersects it.
- Graph several parabolas changing only one of the three constants to see what affect each one has on the position and shape of the parabola.
- If you mix up the x and y axes or the x and y variables, the parabolas will be horizontal instead of vertical.
About the Author
Bert Markgraf is a freelance writer with a strong science and engineering background. He has written for scientific publications such as the HVDC Newsletter and the Energy and Automation Journal. Online he has written extensively on science-related topics in math, physics, chemistry and biology and has been published on sites such as Digital Landing and Reference.com He holds a Bachelor of Science degree from McGill University.