Question

10. Factoring is an algebraic process, which is an essential one in the field of mathematics. Consider the following equation:
𝑦 = 𝑥2 + 2𝑥 − 8
This equation in factored form can be written as:
𝑦 = (𝑥 − 2)(𝑥 + 4)
Explain using words and algebra, how the zeros, the axis of symmetry, and the
vertex can be obtained.

Answer 1

The zeros of an equation are the values of x that make y equal to zero. In other words, they are the values of x where the graph of the equation intersects the x-axis. To find the zeros of the equation 𝑦 = (𝑥 − 2)(𝑥 + 4), we can set y equal to zero and solve for x:

0 = (𝑥 − 2)(𝑥 + 4)

This equation is true if and only if either 𝑥 − 2 = 0 or 𝑥 + 4 = 0. Solving each equation gives us:

𝑥 − 2 = 0 → 𝑥 = 2
𝑥 + 4 = 0 → 𝑥 = −4

Therefore, the zeros of the equation are x = 2 and x = −4.

The axis of symmetry of the graph is a vertical line that divides the graph into two symmetric halves. For a quadratic equation in standard form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, the axis of symmetry is the line given by x = −𝑏/2𝑎. In the case of 𝑦 = (𝑥 − 2)(𝑥 + 4), we can rewrite it in standard form:

𝑦 = 𝑥2 + 2𝑥 − 8

From here, we can see that 𝑎 = 1, 𝑏 = 2, and 𝑐 = −8. Therefore, the axis of symmetry is:

x = −𝑏/2𝑎 = −2/2(1) = −1

The vertex of the graph is the point where the curve reaches its highest or lowest point. For a quadratic equation in standard form, the vertex is given by the point (−𝑏/2𝑎, 𝑐−𝑏2/4𝑎). In the case of 𝑦 = (𝑥 − 2)(𝑥 + 4), we can rewrite it in standard form:

𝑦 = 𝑥2 + 2𝑥 − 8

From here, we can see that 𝑎 = 1, 𝑏 = 2, and 𝑐 = −8. Therefore, the vertex is:

( −𝑏/2𝑎, 𝑐−𝑏2/4𝑎 ) = ( −1, −9 )

Therefore, the zeros of the equation are x = 2 and x = −4, the axis of symmetry is x = −1, and the vertex is (−1, −9).