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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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Example

Figure 1 shows the graph of a quadratic function. Find the formula for this quadratic function and express the formula in vertex form and in standard form.

Figure 1: Find the formula of this quadratic function.

Solution

We will find the formula in vertex form (this is relatively easy as the x- and y-coordinates of the vertex are given) and then convert the vertex form to standard form. The vertex form of a quadratic function has the format:

y = a Â· (x - h)+ k,

where the letter h represents the x-coordinate of the vertex and the letter k represents the y-coordinate of the vertex.

Figure 1 shows that the x-coordinate of the vertex is equal to 3 and that the y-coordinate of the vertex is equal to 1. This means that the vertex form of this quadratic will be:

y = a Â· (x - 3) 2 +1.

All that remains is to find the numerical value of the constant a. To do this, you can use the x- and y-coordinates of any other point (i.e. other than the vertex) that lies on the quadratic â€“ for example the point (0, 4) shown in Figure 3. To work out the value of a we will plug x = 0 and y = 4 into the vertex form and then solve for a.

4 = a Â· (0 - 3) 2 +1.

4 = a Â· 9 +1.

3 = a Â· 9.

So, the equation for the quadratic function shown in Figure 1 (expressed in vertex form) is:

To convert this equation from vertex form to standard form, you can expand by FOILing and then collect like terms.

(Expand the (x â€“ 3)2 by FOILing)

(Multiply through by one third)

(Combine the like terms)

So, the equation for the quadratic function shown in Figure 1 (expressed in standard form) is: