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However students memorize the formula, I like them to also think of the version where dividing by '2a' appears twice: -b/2a + or - (Sq rt of (b² – 4ac))/2a. This gives meaning to the formula: the first term (-b/2a) is the x-value of the axis of the parabola; + or – a positive second term is the distance on either side of that axis to the parabola itself.

Given an example of a quadratic equation, y = ax² + bx + c, I want students to use it to visualize the parabola:

With x = 0, y = c:  'c' is the parabola's y-intercept.

Y = bx + c is a line through c that is easy to visualize ('c' is y-intercept, slope is 'b'). It is tangent to the parabola at c.

'ax²' is the curving factor that applies to the y = bx + c line: it makes the line curb back on itself changing it into the parabola.  

If we cut out a paper parabola for a = 1, for instance, it is easy to position it on a graph tangent to line y = bx + c. This immediately tells us where the roots would be (if there are real roots). It immediately shows that, if 'a' and 'c' are positive, any real root must both be negative. If 'a' is positive and 'b' negative, we immediately know that both roots, if real, must be positive. Just by looking at the equation, students can learn to visualize the graph of the parabola and immediately see many of its essential characteristics.

However students memorize the formula, I like them to also think of the version where dividing by '2a' appears twice: -b/2a + or - (Sq rt of (b² – 4ac))/2a. This gives meaning to the formula: the first term (-b/2a) is the x-value of the axis of the parabola; + or – a positive second term is the distance on either side of that axis to the parabola itself.

Given an example of a quadratic equation, y = ax² + bx + c, I want students to use it to visualize the parabola:

With x = 0, y = c:  'c' is the parabola's y-intercept.

Y = bx + c is a line through c that is easy to visualize ('c' is y-intercept, slope is 'b'). It is tangent to the parabola at c.

'ax²' is the curving factor that applies to the y = bx + c line: it makes the line curb back on itself changing it into the parabola.  

If we cut out a paper parabola for a = 1, for instance, it is easy to position it on a graph tangent to line y = bx + c. This immediately tells us where the roots would be (if there are real roots). It immediately shows that, if 'a' and 'c' are positive, any real root must both be negative. If 'a' is positive and 'b' negative, we immediately know that both roots, if real, must be positive. Just by looking at the equation, students can learn to visualize the graph of the parabola and immediately see many of its essential characteristics.

However students memorize the formula, I like them to also think of the version where dividing by '2a' appears twice: -b/2a + or - (Sq rt of (b² – 4ac))/2a. This gives meaning to the formula: the first term (-b/2a) is the x-value of the axis of the parabola; + or – a positive second term is the distance on either side of that axis to the parabola itself.

Given an example of a quadratic equation, y = ax² + bx + c, I want students to use it to visualize the parabola:

With x = 0, y = c:  'c' is the parabola's y-intercept.

Y = bx + c is a line through c that is easy to visualize ('c' is y-intercept, slope is 'b'). It is tangent to the parabola at c.

'ax²' is the curving factor that applies to the y = bx + c line: it makes the line curb back on itself changing it into the parabola.  

If we cut out a paper parabola for a = 1, for instance, it is easy to position it on a graph tangent to line y = bx + c. This immediately tells us where the roots would be (if there are real roots). It immediately shows that, if 'a' and 'c' are positive, any real root must both be negative. If 'a' is positive and 'b' negative, we immediately know that both roots, if real, must be positive. Just by looking at the equation, students can learn to visualize the graph of the parabola and immediately see many of its essential characteristics.

However students memorize the formula, I like them to also think of the version where dividing by '2a' appears twice: -b/2a + or - (Sq rt of (b² – 4ac))/2a. This gives meaning to the formula: the first term (-b/2a) is the x-value of the axis of the parabola; + or – a positive second term is the distance on either side of that axis to the parabola itself.

Given an example of a quadratic equation, y = ax² + bx + c, I want students to use it to visualize the parabola:

With x = 0, y = c:  'c' is the parabola's y-intercept.

Y = bx + c is a line through c that is easy to visualize ('c' is y-intercept, slope is 'b'). It is tangent to the parabola at c.

'ax²' is the curving factor that applies to the y = bx + c line: it makes the line curb back on itself changing it into the parabola.  

If we cut out a paper parabola for a = 1, for instance, it is easy to position it on a graph tangent to line y = bx + c. This immediately tells us where the roots would be (if there are real roots). It immediately shows that, if 'a' and 'c' are positive, any real root must both be negative. If 'a' is positive and 'b' negative, we immediately know that both roots, if real, must be positive. Just by looking at the equation, students can learn to visualize the graph of the parabola and immediately see many of its essential characteristics.

However students memorize the formula, I like them to also think of the version where dividing by '2a' appears twice: -b/2a + or - (Sq rt of (b² – 4ac))/2a. This gives meaning to the formula: the first term (-b/2a) is the x-value of the axis of the parabola; + or – a positive second term is the distance on either side of that axis to the parabola itself.

Given an example of a quadratic equation, y = ax² + bx + c, I want students to use it to visualize the parabola:

With x = 0, y = c:  'c' is the parabola's y-intercept.

Y = bx + c is a line through c that is easy to visualize ('c' is y-intercept, slope is 'b'). It is tangent to the parabola at c.

'ax²' is the curving factor that applies to the y = bx + c line: it makes the line curb back on itself changing it into the parabola.  

If we cut out a paper parabola for a = 1, for instance, it is easy to position it on a graph tangent to line y = bx + c. This immediately tells us where the roots would be (if there are real roots). It immediately shows that, if 'a' and 'c' are positive, any real root must both be negative. If 'a' is positive and 'b' negative, we immediately know that both roots, if real, must be positive. Just by looking at the equation, students can learn to visualize the graph of the parabola and immediately see many of its essential characteristics.

However students memorize the formula, I like them to also think of the version where dividing by '2a' appears twice: -b/2a + or - (Sq rt of (b² – 4ac))/2a. This gives meaning to the formula: the first term (-b/2a) is the x-value of the axis of the parabola; + or – a positive second term is the distance on either side of that axis to the parabola itself.

Given an example of a quadratic equation, y = ax² + bx + c, I want students to use it to visualize the parabola:

With x = 0, y = c:  'c' is the parabola's y-intercept.

Y = bx + c is a line through c that is easy to visualize ('c' is y-intercept, slope is 'b'). It is tangent to the parabola at c.

'ax²' is the curving factor that applies to the y = bx + c line: it makes the line curb back on itself changing it into the parabola.  

If we cut out a paper parabola for a = 1, for instance, it is easy to position it on a graph tangent to line y = bx + c. This immediately tells us where the roots would be (if there are real roots). It immediately shows that, if 'a' and 'c' are positive, any real root must both be negative. If 'a' is positive and 'b' negative, we immediately know that both roots, if real, must be positive. Just by looking at the equation, students can learn to visualize the graph of the parabola and immediately see many of its essential characteristics.

However students memorize the formula, I like them to also think of the version where dividing by '2a' appears twice: -b/2a + or - (Sq rt of (b² – 4ac))/2a. This gives meaning to the formula: the first term (-b/2a) is the x-value of the axis of the parabola; + or – a positive second term is the distance on either side of that axis to the parabola itself.

Given an example of a quadratic equation, y = ax² + bx + c, I want students to use it to visualize the parabola:

With x = 0, y = c:  'c' is the parabola's y-intercept.

Y = bx + c is a line through c that is easy to visualize ('c' is y-intercept, slope is 'b'). It is tangent to the parabola at c.

'ax²' is the curving factor that applies to the y = bx + c line: it makes the line curb back on itself changing it into the parabola.  

If we cut out a paper parabola for a = 1, for instance, it is easy to position it on a graph tangent to line y = bx + c. This immediately tells us where the roots would be (if there are real roots). It immediately shows that, if 'a' and 'c' are positive, any real root must both be negative. If 'a' is positive and 'b' negative, we immediately know that both roots, if real, must be positive. Just by looking at the equation, students can learn to visualize the graph of the parabola and immediately see many of its essential characteristics.