Q54. - Algebra 2

Question

Write an expression for the minimum value of a function of the form $y=a{x}^2+c$, where $a>0$. Explain your reasoning. Then use this function to find the minimum value of $y=8.6x^2-12.5$

Step-by-Step Solution

Answer

The minimum value of a function of the form $y=a{x}^2+c$ is c. Using the function, the minimum value of $y=8.6x^2-12.5$ is -12.5.

Step 1. Given Information.

Given to write an expression for the minimum value of a function of the form $y=a{x}^2+c$, where $a>0$.

Then using the function, the minimum value of $y=8.6x^2-12.5$ is to be determined

Step 2. Explanation .

When a is positive, the function $y=a{x}^2+c$ represents an upwards opening parabola and has the minimum value at the vertex.

Vertex of a parabola lies on the axis of symmetry.

For an equation of the form $f\left(x\right)=a{x}^2+bx+c$, the axis of symmetry is given by $x=-\frac{b}{2a}$

Here for the given equation, b=0

Plugging the values in the equation:

 $\begin{array}{l}x=-\frac{b}{2a}\\ x=-\frac{0}{2a}\\ x=0\end{array}$

Hence the axis of symmetry is x=0.

 

The x-coordinate of the vertex is same as the axis of symmetry.

So, the y-coordinate of the vertex can be obtained by plugging the value in the equation.

Plugging x=0 in the equation:

 $\begin{array}{l}y=a{x}^2+c\\ y=a{\left(0\right)}^2+c\\ y=0+c\\ y=c\end{array}$

Hence the coordinate of the vertex is $\left(0,c\right)$.

Therefore, the minimum value of the function is c.

 

Using this rule, the minimum value of the function $y=8.6x^2-12.5$ is given by:

$c=-12.5$

Step 3. Conclusion .

The minimum value of a function of the form $y=a{x}^2+c$ is c. Using the function, the minimum value of $y=8.6x^2-12.5$ is -12.5.