How can you use the values being subtracted from x and y to adjust where the circle is?

For an interactive conceptual version of this problem created in GeoGebra, click here.

There are many potential answers but all will come from a circle with a radius of 5 units including:

(x – -2)^2 + (y – -1)^2 = 5^2

Point on circle: (-6, -4)

(x – -2)^2 + (y – -1)^2 = 5^2

Point on circle: (-6, -4)

(x – -2)^2 + (y – -1)^2 = 5^2

Point on circle: (1, 3)

Source: Robert Kaplinsky

The post Equations of Circles 1 first appeared on Open Middle®.]]>How can you use the values being subtracted from x and y to adjust where the circle is?

There are many potential answers but all will come from a circle with a radius of 5 units. The closest point to the origin found so far is 2 units away. Here’s one example:

(x – -5)^2 + (y – -4)^2 = 5^2

Point on circle: (-2, 0)

(x – -5)^2 + (y – -4)^2 = 5^2

Point on circle: (-2, 0)

Source: Robert Kaplinsky

The post Equations of Circles 2 first appeared on Open Middle®.]]>How do each of the values you can choose affect the values of the roots and range?

There are many possible answers but all will have a minimum and point upwards including:

y = 1x^2 + 6x + 8

roots: x = -4 and x = -2

range: y ≥ -1

y = 1x^2 + 6x + 5

roots: x = -5 and x = -1

range: y ≥ -4

Source: Robert Kaplinsky

The post Trinomial Function Features 1 first appeared on Open Middle®.]]>How do each of the values you can choose affect the values of the roots and range?

y = 1x^2 + 6x + 8

roots: x = -4 and x = -2

range: y ≥ -1

Have you found an equations with roots that are even closer together? Share it in the comments.

Source: Robert Kaplinsky

The post Trinomial Function Features 2 first appeared on Open Middle®.]]>How do each of the values you can choose affect the values of the roots? Note that range is being defined as the difference between the least and greatest root values.

The least range possible that has been found so far is 4 and comes from the choices below. If you find a better one, please share it in the comments.

y = 1x^3 + -4x^2 + 3x + 0

roots: x = 0, x = 1, and x = 4

Source: Robert Kaplinsky

The post Polynomial Function Features 2 first appeared on Open Middle®.]]>How do each of the values you can choose affect the values of the roots?

There are multiple answers including:

y = 1x^3 + -4x^2 + 3x + 0

roots: x = 0, x = 1, and x = 4

y = -1x^3 + -3x^2 + 4x + 0

roots: x = -4, x = 0, and x = 1

Source: Robert Kaplinsky

The post Polynomial Function Features 1 first appeared on Open Middle®.]]>How do each of the values you can choose affect the values of the y-intercept?

The greatest possible y-intercept is 9 and can be gotten from an equation like:

y = 4 * log2(x – (-8)) + -3

y-intercept: 9

Source: Robert Kaplinsky

The post Logarithmic Function Features 2 first appeared on Open Middle®.]]>How do each of the values you can choose affect the values of the y-intercept?

There are many possible answers including:

y = 4 * log2(x – (-8)) + -6

y-intercept: 6

Source: Robert Kaplinsky

The post Logarithmic Function Features 1 first appeared on Open Middle®.]]>How do each of the values you can choose affect the values of the y-intercept?

The greatest possible y-intercept is 9 and can be gotten through an equation like:

y = 4 * 2^(x+0) + 5

y-intercept: 9

Source: Robert Kaplinsky

The post Exponential Function Features 2 first appeared on Open Middle®.]]>How do each of the values you can choose affect the values of the y-intercept?

There are many possible answers including:

y = 4 * 2^(x+0) + -6

y-intercept: -2

Source: Robert Kaplinsky

The post Exponential Function Features 1 first appeared on Open Middle®.]]>