Identifying the Vertex

Find the vertex of $y = 2(x - 3)^2 + 5$

Identify the form: This is in vertex form: $y = a(x - h)^2 + k$ = $a = 2$, $h = 3$, $k = 5$

Find h (x-coordinate): In $(x - h)^2$, we have $(x - 3)^2$, so $h = 3$ = $h = 3$

Find k (y-coordinate): The constant at the end is $k = 5$ = $k = 5$

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

Answer: The vertex is $(3, 5)$. Since $a = 2 > 0$, the parabola opens upward, so $(3, 5)$ is the minimum point.

Handling Negative Signs

Find the vertex of $y = -(x + 4)^2 - 2$

Rewrite to match vertex form: $y = -(x + 4)^2 - 2 = -1(x - (-4))^2 + (-2)$ = Vertex form identified

Identify h: $(x + 4) = (x - (-4))$, so $h = -4$ = $h = -4$

Identify k: The constant is $-2$, so $k = -2$ = $k = -2$

State the vertex: Vertex = $(h, k) = (-4, -2)$ = $(-4, -2)$

Answer: The vertex is $(-4, -2)$. Since $a = -1 < 0$, the parabola opens downward, so $(-4, -2)$ is the maximum point.

Writing Vertex Form from a Graph

A parabola has vertex $(2, -3)$, opens upward, and passes through $(4, 5)$. Write its equation in vertex form.

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

Substitute the vertex: $h = 2$, $k = -3$: $y = a(x - 2)^2 - 3$ = $y = a(x - 2)^2 - 3$

Use the point to find a: Substitute $(4, 5)$: $5 = a(4 - 2)^2 - 3$ = $5 = a(2)^2 - 3$

Solve for a: $5 = 4a - 3 \Rightarrow 8 = 4a \Rightarrow a = 2$ = $a = 2$

Write final equation: $y = 2(x - 2)^2 - 3$ = Complete equation

Answer: $y = 2(x - 2)^2 - 3$

Finding the Axis of Symmetry

Find the axis of symmetry and direction of $y = -3(x - 1)^2 + 7$

Identify h: From $(x - 1)^2$, we get $h = 1$ = $h = 1$

Write axis of symmetry: Axis of symmetry is $x = h$ = $x = 1$

Determine direction: $a = -3 < 0$, so parabola opens downward = Opens downward

Interpret the vertex: Vertex $(1, 7)$ is the maximum point = Maximum at $(1, 7)$

Answer: Axis of symmetry: $x = 1$. The parabola opens downward with maximum point $(1, 7)$.

Mistake: Getting the sign of h wrong: thinking $(x + 4)^2$ means $h = 4$

Why: The formula is $(x - h)^2$. When you see $(x + 4)^2$, it's actually $(x - (-4))^2$.

Correct: If you see $+$ inside the parentheses, $h$ is negative. $(x + 4)^2$ means $h = -4$.

Mistake: Confusing which direction the parabola opens

Why: Students sometimes think $a$ affects horizontal direction instead of vertical.

Correct: $a > 0$ means opens UP (like a smile). $a < 0$ means opens DOWN (like a frown).

Mistake: Forgetting that the vertex is a minimum when $a > 0$ and maximum when $a < 0$

Why: The vertex is always an extreme point, but which type depends on the direction.

Correct: Opening up = valley = minimum. Opening down = hill = maximum.

Projectile Motion

When an object is thrown, its height over time follows a parabola. Vertex form reveals the maximum height.

A ball's height is $h(t) = -5(t - 2)^2 + 25$ meters. The vertex $(2, 25)$ tells us the ball reaches 25 meters at $t = 2$ seconds.

Business Optimization

Companies use quadratic functions to model profit. The vertex shows the price that maximizes profit.

Profit function $P(x) = -2(x - 50)^2 + 5000$ has vertex $(50, 5000)$. Selling at 50 dollars gives maximum profit of 5000 dollars.

Architecture - Parabolic Arches

Architects design arches using parabolas. The vertex determines the highest point of the arch.

An arch modeled by $y = -0.1(x - 10)^2 + 10$ has its peak at $(10, 10)$ - the center is 10 meters high.

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

A: Use completing the square. For $y = ax^2 + bx + c$, factor out $a$ from the first two terms, complete the square, then simplify. Alternatively, find the vertex using $h = -\frac{b}{2a}$ and $k = f(h)$.

Q: Can a be a fraction or decimal?

A: Yes! If $|a| < 1$ (like $a = 0.5$ or $a = \frac{1}{2}$), the parabola is wider. If $|a| > 1$, it's narrower.

Q: What if there's no number in front of the parentheses?

A: Then $a = 1$. For example, $y = (x - 3)^2 + 2$ has $a = 1$.