What is the vertex of the parabola y = 3(x − 4)² + 7?
A(−4, 7) — reading the number inside the parentheses as −4
B(4, 7) — because x must equal 4 to make (x − 4)² equal zero
C(4, −7) — negating k to find the vertex y-coordinate
D(−4, −7) — negating both h and k
In y = a(x − h)² + k, the vertex is (h, k). Here h = 4 and k = 7, so the vertex is (4, 7). The most common error is reading h as −4 because of the minus sign in (x − 4)². The minus sign is built into the form's structure: you need x = 4 (not x = −4) to make (x − 4)² equal zero, so the vertex is at x = +4.
Question 2 Multiple Choice
Compared to the parent function y = x², how does the parabola y = (1/3)(x − 2)² + 1 differ in width?
ANarrower — because the coefficient 1/3 is smaller, making it more compressed
BWider — because |a| = 1/3 < 1 causes vertical compression that spreads the parabola outward
CSame width — only shifted right 2 and up 1
DNarrower and opens downward because of the fraction
When |a| < 1, the parabola is wider (flatter) than y = x². When |a| > 1, it is narrower. The coefficient 1/3 compresses the vertical scale, which visually spreads the parabola outward. The most common misconception is reversing this relationship: thinking a smaller coefficient makes a narrower graph because 'smaller = less.' Think of it instead as: a larger a value pulls the sides of the parabola inward (steeper), while a smaller a lets them spread out (flatter).
Question 3 True / False
In the equation y = (x + 3)² + 1, the vertex is at (3, 1) because the number inside the parentheses is 3.
TTrue
FFalse
Answer: False
The vertex is at (−3, 1). Rewriting in standard vertex form: y = (x + 3)² + 1 = (x − (−3))² + 1, so h = −3 and k = 1. The vertex is where the squared expression equals zero: x + 3 = 0 gives x = −3. The sign trap works in both directions — when you see (x + 3)², the vertex is at x = −3, not x = +3.
Question 4 True / False
The vertex form y = a(x − h)² + k and the standard form y = ax² + bx + c represent the same quadratic function — they are algebraically equivalent, just written differently.
TTrue
FFalse
Answer: True
They are identical functions. Vertex form can be expanded to standard form by multiplying out (x − h)², distributing a, and collecting constants. Standard form can be converted to vertex form by completing the square. The same parabola, the same a value, the same vertex — just different algebraic presentations that make different information immediately visible.
Question 5 Short Answer
In y = a(x − h)² + k, why does the graph shift RIGHT by h units rather than LEFT when h is positive? Many students expect the direction to match the sign shown.
Think about your answer, then reveal below.
Model answer: The vertex is located where the squared expression equals zero — the minimum (or maximum) of the function. For (x − h)², this happens when x = h: substituting x = h gives (h − h)² = 0. So the vertex is at x = h, which means a positive h shifts the parabola to the right. The minus sign in the form (x − h)² is structural: to place the vertex at x = 3, you write (x − 3)², not (x + 3)². The direction of the shift is opposite to what the sign suggests because the vertex must satisfy the equation, not be read directly from the sign.
This sign convention is the most persistent source of error in vertex form. One reliable check: ask 'what value of x makes the expression inside the parentheses equal zero?' That value is always the x-coordinate of the vertex, regardless of what signs appear in the written form.