Let's investigate ways to use a table of values to represent the solution to a quadratic equation.

TEKS Standards and Student Expectations

A(8) Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to:

A(8)(B) solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

A(8)(A) write, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems

Resource Objective(s)

Use a table of values and a given graph to find the solution to a quadratic equation.

Essential Questions

How can the solution(s) to a quadratic equation be found from a table of values?

How can the solution(s) to a quadratic equation be found from a graph?

How can the solution(s) to a quadratic equation be estimated from a table?

Vocabulary

• Equation
• Function
• x-Intercept
• Parabola 
• Quadratic Equation
• Zero

A table and a graph can both be used to show solutions to a quadratic equation.

The graph and table below show points for the quadratic function

$y = x^2- x - 6.$

Both representations of a quadratic equation can be used to find the solution. 

The solutions to quadratic equations are called roots. Roots are the x-intercepts (zeros ) of a quadratic function. 

For every quadratic equation, there is a related quadratic function. For example, if you are given the quadratic equation

$x^2 + 5x + 4 = 0$,

the related quadratic function is 

$f\left(x\right) = x^2 + 5x + 4.$

A quadratic equation may have two solutions, one solution, or no solution. 

Answer the following questions about quadratic equations and functions.

A table of values can be generated from a quadratic function by substituting the x-values and calculating the values for f(x).

When looking at a table of values for a quadratic function, the x-intercepts represent the x-values where y = 0. This corresponds to the x-values where f(x) is 0 in function notation.

Substitution can be used to verify the solutions to the function 

$f\left(x\right)=x^2-x-6$. 

Sometimes when you have a table of values for a function, the solutions to the related equation are not obvious. For example, consider the function

$y = x^2 - 2$

and its related equation

$x^2 - 2 = 0$.

The table below displays the relationship from the equation.

If you want to solve the related equation

$x^2 - 2 = 0$

from this table, there are no y-values equal to 0 to use. In this case, we must estimate where the zeros are from the table. 

Click on the image below to view a video on estimating the zeros of a quadratic function when given a table. As you watch, think about how you can use the table above to estimate the solutions to the equation

 $x^{2 }- 2 = 0.$

You have investigated different ways to determine the solutions to quadratic equations using tables of values.

In doing so, you also compared functions to their related equations.

In a table of values, solutions to related equations can be found by locating rows containing ordered pairs where the function value, or y-value, is equal to 0. In some cases, the solution must be estimated.