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Quadratic Equations (Parabolas)

Page history last edited by Andrea Grieser 13 years, 8 months ago

Quadratic equations are second degree polynomialsParabolas (Quadratic Functions)

The shape formed by quadratics are called parabolas.  We see parabolas in nature when we look at tractories (throwing a ball or firing a missile), in the way a cable hangs on a suspension bridge, and in many physics applications.

 

They are often shown in their standard form of y = ax2 + bx + c, where a, b, and c are real numbers.  They also may appear in their vertex form of y = a(x - h)2 + k, where (h, k) represent the coordinates of the parabola's vertex.

 

This spreadsheet allows you to experiment with different values of a, b, and c.

 

Quadratic equations of this form are functions.  A parabola has an axis of symmetry, which is a line that is not part of the function, but around which the function is symmetric (there are matching points on each side of the line.  The vertex of a parabola is the turning point; this is where the curve changes from falling to rising (if we have a concave up ("smile") parabola) or from rising to falling (if we have a concave down ("frowning") parabola).

 

 

We can determine if a parabola will be concave up, if the a constant in ax2 + bx + c is positive.  If the a constant is negative, the parabola will be concave down.  The vertex of a parabola is the minimum (has the smallest y-value) when the parabola is concave up.  The vertex is a maximum (has the largest y-value) when the parabola is concave down.

 

We can find the axis of symmetry by putting the quadratic in standard form, ax2 + bx + c, and setting x = -b/2a.  Remember that the line formed by x = constant is a vertical line, NOT a single point.

 

We can sketch a parabola by plotting the y-intercept (the c constant in ax2 + bx + c).  We can then lightly sketch the axis of symmetry by graphing x = -b/2a.   We can find the vertex by finding the corresponding y-value that matches x = -b/2a, because the vertex is always on the axis of symmetry, so we use it as our x-value (plug this value back into ax2 + bx + c to get the corresponding y-value).  Then we can pick some x-values and calculate the y-value, remembering that since the curve is symmetric around the axis of symmetry, finding the y-value for one side finds the corresponding y-value for the other side of the axis of symmetry. 

 

We can also graph quadratic equations using graphing calculators.  When using a TI-84 type calculator, pressing the Y= button will allow us to type in the equation so that it can be graphed.

 

We are sometimes asked to find the zeros of a quadratic equation.  This means we want to find the x-intercepts of the graph of the quadratic, where the curve touches or crosses the x-axis.   Zeros are also called the roots of the quadratic equation.  Since the zeros are x-intercepts, the y value of the x-intercept will be 0 (since that is true of all x-intercepts).  So we would set the quadratic y = ax2 + bx + c equal to 0:  ax2 + bx + c = 0.  

 

There are a number of approaches to solving ax2 + bx + c = 0.  By solving, we mean find the value(s) of x, if any, that make the quadratic equation true.  We can graph the quadratic, either by hand or on a graphing calculator, and note where the curve crosses or touches the x-axis.  We can use the quadratic formula.  We can also factor the quadratic, and use the zero product property to find the zeros.  We can determine the number of zeros by using the discriminant.  This article explains solving quadratics in more detail.

 

These class notes provide further explanation and examples. 

 

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