Question

The vertex of the parabola $y=(x-2)(x-8)+7$ is

• A. (5, 2)
• B. $(5,-2)$
• C. $(-5,-2)$
• D. $(-5,2)$
• E. (2, 8)

Hint:

Shortcut Tip: The x-coordinate of the vertex is always exactly halfway between the x-intercepts of the factored part. The roots of $(x-2)(x-8)$ are 2 and 8. The midpoint is $(2+8)/2 = 5$. Plug in $x=5$ to get $y=(3)(-3)+7 = -2$!

Answer 1

The Correct Option is B

Concept:
To find the vertex of a parabola given in factored or irregular form, it is often easiest to expand the equation into the standard quadratic form $y = ax^2 + bx + c$. From there, the x-coordinate of the vertex is located exactly at the axis of symmetry: $x = -\frac{b}{2a}$.

Step 1: Expand the binomial product.
Multiply the terms $(x-2)$ and $(x-8)$: $$y = (x^2 - 8x - 2x + 16) + 7$$ $$y = x^2 - 10x + 16 + 7$$

Step 2: Combine constants to get standard form.
Add the constant terms together: $$y = x^2 - 10x + 23$$ Now the equation is in the standard $ax^2 + bx + c$ form with $a = 1$, $b = -10$, and $c = 23$.

Step 3: Find the x-coordinate of the vertex.
Use the vertex formula $x = -\frac{b}{2a}$: $$x = -\frac{-10}{2(1)}$$ $$x = \frac{10}{2} = 5$$

Step 4: Find the y-coordinate of the vertex.
Substitute $x = 5$ back into the standard form equation to find the corresponding y-value: $$y = (5)^2 - 10(5) + 23$$ $$y = 25 - 50 + 23$$

Step 5: Calculate the final coordinates.
Perform the final arithmetic addition: $$y = -25 + 23 = -2$$ The vertex $(x, y)$ is $(5, -2)$. Hence the correct answer is (B) $(5,-2)$.