Explore: Writing Quadratic Functions Given Three Points Solutions

Writing Quadratic Functions Given Three Points Solutions

You can write the equation for a quadratic function if you have three points on a parabola.
However, because you do not know if any of the points is the vertex , you cannot write the equation in vertex form.
Instead, you can use a method that combines:
- solving a system of equations with three variables and
- working with the standard form of a quadratic equation, using either $y = a x^{2} + b x + c$ or $x = a y^{2} + b y + c .$
To confirm that your equation is correct, you can substitute each point into the equation, or utilize technology to graph the equation and confirm that all three points are on the parabola.

You need to eliminate at least one variable in two out of the three equations to begin the process of solving. Then solve for one of the remaining variables and use repeated substitution to find all of the variables and check your work.

Example 6

Write the equation of a parabola in $x = a y^{2} + b y + c$ form using the points $D (14, 1), E (7, - 2.5), F (10, 0.5)$ .

Implement

$x = a y^{2} + b y + c \Leftrightarrow a y^{2} + b y + c = x$

$D (14, 1) a {(1)}^{2} + b (1) + c = 14 D : a + b + c = 14$

$E (7, - 2.5) a {(- 2.5)}^{2} + b (- 2.5) + c = 7 E : 6.25 a - 2.5 b + c = 7$

$F (10, 0.5) a {(0.5)}^{2} + b (0.5) + c = 10 F : 0.25 a + 0.5 b + c = 10$

$\begin{array}{rcl} - 6.25 a + 2.5 b - c & = & - 7 \\ + a + b + c = 14 \\ - 5.25 a + 3.5 b & = & 7 \end{array}$

$6.25 a - 2.5 b + c = 7 + - 0.25 a - 0.5 b - c = - 10 6 a - 3 b = - 3$

$(- 5.25 a + 3.5 b = 7) (3) \Rightarrow - 15.75 a + 10.5 b = 21 (6 a - 3 b = - 3) (3.5) \Rightarrow + 21 a - 10.5 b = - 10.5 5.25 a = 10.5 a = 2$

$\begin{array}{rcl} \begin{matrix} 6 a - 3 b = - 3 & a + b + c = 14 \\ 6 (2) - 3 b = - 1 & 2 + 5 + c = 14 \\ - 3 b = - 15 & c = 7 \\ b = 5 \end{matrix} \end{array}$

$x = 2 y^{2} + 5 y + 7$

Explain

Eliminate c from the first two equations $(D + - E)$
Eliminate c from the last two equations $(E + - F)$  
Eliminate b, solve for a  
Solve for b, solve for c 
Write the equation

Example 7

Write the equation of the parabola in $y = a x^{2} + b x + c$ form using the points $(1, 5.5), (- 2, 19), (3, 1.5)$ .

$(1, 5.5) a {(1)}^{2} + b (1) + c = 5.5 a + b + c = 5.5$

$(- 2, 19) a {(- 2)}^{2} + b (- 2) + c = 19 4 a - 2 b + c = 19$

$(3, 1.5) a {(3)}^{2} + b (3) + c = 1.5 9 a + 3 b + c = 1.5$

$4 a - 2 b + c = 19 + - a - b - c = - 5.5 3 a - 3 b = 13.5$

$4 a - 2 b + c = 19 + - 9 a - 3 b - c = - 1.5 - 5 a - 5 b = 17.5$

$(3 a - 3 b = 13.5) (5) \Rightarrow 15 a - 15 b = 67.5 (- 5 a - 5 b = 17.5) (- 3) \Rightarrow 15 a + 15 b = - 52.5 30 a = 15 a = 0.5 3 (0.5) - 3 b = 13.5 1.5 - 3 b = 13.5 - 3 b = 12 b = - 4 a + b + c = 5.5 (0.5) + (- 4) + c = 5.5 - 3.5 + c = 5.5 c = 9 y = 0.5 x^{2} - 4 x + 9$