# How do you write a quadratic function in standard form whose graph passes through points (1/2, -3/10), (1, 6/5), (1/4, -3/10)?

Apr 18, 2017

Start with the vertex form and use symmetry to find the x coordinate.
Use the points to find the other 2 two values of the vertex form.
Expand the vertex form into the standard form.

#### Explanation:

Because you specified a function we discard the vertex form $x = a {\left(y - k\right)}^{2} + h$ and use only the form:

$y = a {\left(x - h\right)}^{2} + k \text{ [1]}$

Because the we know the 2 x values corresponding to y value, $- \frac{3}{10}$ we know know that the x coordinate of the vertex is halfway between $\frac{1}{4} \mathmr{and} \frac{1}{2}$

$h = \frac{\frac{1}{2} + \frac{1}{4}}{2}$

$h = \frac{3}{8}$

Substitute $\frac{3}{8}$ for h into equation [1]:

$y = a {\left(x - \frac{3}{8}\right)}^{2} + k \text{ [2]}$

Substitute the point $\left(\frac{1}{2} , - \frac{3}{10}\right)$ into equation [2]:

$- \frac{3}{10} = a {\left(\frac{1}{2} - \frac{3}{8}\right)}^{2} + k$

$- \frac{3}{10} = a {\left(\frac{1}{8}\right)}^{2} + k$

$- \frac{3}{10} = \frac{a}{64} + k \text{ [3]}$

Substitute the point $\left(1 , \frac{6}{5}\right)$ into equation [2]:

$\frac{6}{5} = a {\left(1 - \frac{3}{8}\right)}^{2} + k$

$\frac{6}{5} = a {\left(\frac{5}{8}\right)}^{2} + k$

$\frac{6}{5} = \frac{25 a}{64} + k \text{ [4]}$

Subtract equation [3] from equation [4]:

$\frac{6}{5} + \frac{3}{10} = \frac{24 a}{64}$

$\frac{15}{10} = \frac{24}{64} a$

$a = \frac{3}{2} \left(\frac{64}{24}\right)$

$a = \frac{1}{2} \left(\frac{64}{8}\right)$

$a = 4$

Use equation [3] to find the value of k:

$- \frac{3}{10} = \frac{4}{64} + k$

$- \frac{3}{10} = \frac{1}{16} + k$

$k = - \frac{3}{10} - \frac{1}{16}$

$k = - \frac{29}{80}$

Substitute the value of "a" and "k" into equation [2]:

$y = 4 {\left(x - \frac{3}{8}\right)}^{2} - \frac{29}{80}$

Expand the square:

$y = 4 \left({x}^{2} - \frac{6}{8} x + \frac{9}{64}\right) - \frac{29}{80}$

Distribute the 4:

$y = 4 {x}^{2} - 3 x + \frac{9}{16} - \frac{29}{80}$

This is the standard form:

$y = 4 {x}^{2} - 3 x + \frac{1}{5}$

The following is a graph of the equation and the 3 points: