Quadratic equations function in the same manner as linear equations in that you have a particular input resulting in a particular output. You plot points on the graph in the same manner as you do linear equations. The difference is that the graph of a quadratic forms a curve, called a parabola, instead of a straight line.
The standard form of a quadratic equation is:
You may also see a quadratic equation written in vertex form. The vertex is where the direction of a parabola changes from negative to positive, or vice versa.
You may be more likely to see vertex form when dealing with transformations of quadratic graphs.
The coefficient a determines the direction and width of the parabola. A larger number creates a skinnier parabola, and a number between -1 and 1 creates a fatter parabola (ignoring signs). Moreover, if a is positive the parabola curves up (positive = smiley face), whereas if a is negative the parabola curves down (negative = sad face).
We know that c is the y-intercept because substituting x = 0 for x² and x gives:
y = a(0)² + b(0) + c = 0 + 0 + c = c
Now, take a look at graphs of quadratic equations:
In graph (a), the coefficient a is positive and the parabola opens up (in a “smile”). In graph (b) the coefficient a is negative and the parabola opens down (in a “frown”).
You will need to know how to find the x-intercepts of a quadratic equation. To find a quadratic equation’s x-intercepts, the quadratic equation must be set equal to zero (0). The reason for this is that for any coordinate that falls on the x-axis, the value of y will be zero (ex. (3, 0) or (‐3, 0)). This is also true when solving for x for quadratic equations, since this technique allows us to simplify our math tools! If your quadratic equation is set to anything other than zero, you must manipulate the quadratic equation to set it equal to zero. Remember that you aren’t going to invent a new way to solve quadratic equations on a standardized test, so either factor or use the quadratic formula once you set the equation equal to 0!
The x-intercepts are also known as the “zeros” or “roots,” since they occur when the function equals zero. We can find the x-intercepts by setting the equation equal to 0 and factoring.
Sometimes, we are unable to factor a quadratic equation. This means that the zeros will either be irrational or imaginary. We can solve for these zeros by using the quadratic formula.
The vertex is the highest or lowest point on a parabola. It is written in the form (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex. If our quadratic equation is not written in vertex form, we can solve for the vertex with the following formula:
Once you have solved for h, you can input this value as the x input in the quadratic equation. The value of y once you solve the equation is the value of k.
Keep in mind, when a quadratic is positive the vertex is the lowest point on the parabola, and when a quadratic is negative the vertex is the highest point on the parabola.
You may be asked to find the equivalent vertex form of an equation given in standard form. We can solve for the vertex and plug it in to the form
To convert an equation from vertex to standard form, we can expand the squared factor and combine like terms.
The easiest way to graph quadratic transformations is to first make sure you have your equation in vertex form, as this form easily demonstrates the different types of transformations.
To transform a quadratic equation, we first start with the parent function, y = x² and transform the graph according to the following rules.
Vieta's Formulas relate the coefficients of a polynomial to the sums and products of its roots.
Given a quadratic equation f(x) = ax² + bx + c with roots p and q found by setting f(x) = 0,
The sum of the roots p + q = -b/a.
The product of the roots p * q = c/a.
The height of a potato thrown in the air can be modeled by the equation . What is the maximum height of the potato?
What is the product of the x-intercepts of ?
The height of a ball thrown in the air off the top of a building can be modeled by the equation , where is time in seconds. If the ball is thrown in the air at time , what time does the ball hit the ground?
Which of the following equations best represents the graph below?
5. Write the equation in standard form.