Identifying Vertex and Direction

For $y = 2(x - 3)^2 - 5$, find the vertex, axis of symmetry, and direction of opening.

Identify $h$ from $(x - h)$: $(x - 3)$ means $h = 3$ = $h = 3$

Identify $k$ as the constant: $k = -5$ = $k = -5$

Write the vertex: Vertex = $(h, k)$ = $(3, -5)$

Find axis of symmetry: $x = h$ = $x = 3$

Determine direction from $a$: $a = 2 > 0$ = Opens upward

Answer: Vertex: $(3, -5)$, Axis of symmetry: $x = 3$, Opens upward (narrower than $y = x^2$)

Writing Vertex Form from a Graph

A parabola has vertex $(-2, 4)$ and passes through the point $(0, 0)$. Write its equation in vertex form.

Start with vertex form: $y = a(x - h)^2 + k$ = Template equation

Substitute vertex $(h, k) = (-2, 4)$: $y = a(x - (-2))^2 + 4$ = $y = a(x + 2)^2 + 4$

Substitute point $(0, 0)$ to find $a$: $0 = a(0 + 2)^2 + 4$ = $0 = 4a + 4$

Solve for $a$: $4a = -4$, so $a = -1$ = $a = -1$

Write final equation: $y = -1(x + 2)^2 + 4$ = $y = -(x + 2)^2 + 4$

Answer: $y = -(x + 2)^2 + 4$

Converting from Standard to Vertex Form

Convert $y = x^2 - 6x + 5$ to vertex form by completing the square.

Group $x$ terms: $y = (x^2 - 6x) + 5$ = Grouped terms

Find value to complete square: $\left(\frac{-6}{2}\right)^2 = (-3)^2 = 9$ = Add and subtract $9$

Add and subtract inside parentheses: $y = (x^2 - 6x + 9 - 9) + 5$ = Balance maintained

Factor perfect square trinomial: $y = (x^2 - 6x + 9) - 9 + 5$ = $y = (x - 3)^2 - 4$

Identify vertex: Comparing to $a(x - h)^2 + k$ = Vertex: $(3, -4)$

Answer: $y = (x - 3)^2 - 4$, with vertex $(3, -4)$

Mistake: Writing vertex as $(h, k)$ when equation has $(x + h)$

Why: When you see $(x + 2)^2$, this is $(x - (-2))^2$, so $h = -2$, not $2$.

Correct: Always rewrite to match $(x - h)$: if you see $(x + 2)$, then $h = -2$.

Mistake: Forgetting that $a$ affects width, not just direction

Why: Students focus only on whether $a$ is positive or negative.

Correct: $|a| > 1$ makes the parabola narrower; $|a| < 1$ makes it wider.

Mistake: Errors in completing the square

Why: Forgetting to subtract the same value that was added inside the parentheses.

Correct: When adding $9$ to complete the square, you must also subtract $9$ to keep the equation balanced.

Projectile Motion

When an object is thrown or launched, its height over time follows a parabolic path. Vertex form immediately tells us the maximum height.

A ball's height is $h(t) = -16(t - 2)^2 + 64$ feet. The vertex $(2, 64)$ tells us the ball reaches 64 feet at $t = 2$ seconds.

Satellite Dishes

Parabolic dishes focus signals at the vertex. Engineers use vertex form to design the precise shape.

A dish with equation $y = 0.25(x - 0)^2 + 0$ has its focal point above the vertex at the origin.

Q: How do I know if the vertex is a maximum or minimum?

A: Look at $a$: if $a > 0$, the parabola opens upward so the vertex is a minimum. If $a < 0$, it opens downward so the vertex is a maximum.

Q: Why is there a negative sign in $(x - h)$?

A: The form $(x - h)$ represents a horizontal shift. When $h = 3$, we write $(x - 3)$, which shifts the parabola 3 units right. When $h = -2$, we write $(x - (-2)) = (x + 2)$, shifting 2 units left.

Q: How do I convert vertex form to standard form?

A: Expand the squared term: $a(x - h)^2 + k = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k$.