Quadratic Functions & Parabola Graphs

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✖️ 1. Identifying vertex, axis of symmetry, and intercepts of a parabola

🎯 Finding Key Points on a Parabola

The vertex is the highest or lowest point on the parabola.
Find the vertex x-coordinate using $x = -\frac{b}{2a}$ , then plug into the equation for y.
The axis of symmetry is the vertical line $x = -\frac{b}{2a}$ that splits the parabola in half.
The y-intercept is found by setting $x = 0$ , giving the point $(0, c)$ .
The x-intercepts are found by solving $ax^2 + bx + c = 0$ using factoring or the quadratic formula.

For $y = 2x^2 - 8x + 6$ : vertex x is $-\frac{-8}{2(2)} = 2$ , so vertex is $(2, -2)$ ; axis is $x = 2$ ; y-intercept is $(0, 6)$ .

💡 The vertex x-value is always halfway between the x-intercepts!

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1. Identifying vertex, axis of symmetry, and intercepts of a parabola

Identifying vertex, axis of symmetry, and intercepts of a parabola

A parabola defined by $y = ax^2 + bx + c$ has three key features: the vertex (turning point), the axis of symmetry (vertical line through the vertex), and intercepts (where the graph crosses the axes). The vertex represents the minimum or maximum point of the function.

Intuition: The vertex is the "tip" of the parabola, the axis of symmetry divides it into mirror halves, and intercepts show where the parabola meets the coordinate axes.

Core Rules:

Vertex: Located at $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$
Axis of symmetry: Vertical line $x = -\frac{b}{2a}$
Y-intercept: Point $(0, c)$ where the parabola crosses the y-axis
X-intercepts: Solutions to $ax^2 + bx + c = 0$ (may be 0, 1, or 2 real solutions)

Consequence: These features completely determine the parabola's position and shape in the coordinate plane.

Example: For $y = 2x^2 - 4x + 1$ , the vertex is at $x = -\frac{-4}{2(2)} = 1$ , giving vertex $(1, -1)$ , axis $x = 1$ , y-intercept $(0, 1)$ .

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Find the x-coordinate of the vertex for the parabola defined by $y = 2x^2 - 8x + 5$ .

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✖️ 2. The effect of coefficient 'a' on the parabola's direction and width (stretching)

🔄 How 'a' Controls the Parabola Shape

If $a > 0$ , the parabola opens upward (smiles).
If $a < 0$ , the parabola opens downward (frowns).
Larger values of $|a|$ make the parabola narrower (steeper sides).
Smaller values of $|a|$ (closer to zero) make the parabola wider (flatter).
The value of $a$ does NOT change the vertex location, only the shape.

Compare $y = 3x^2$ (narrow, opens up) with $y = 0.5x^2$ (wide, opens up) and $y = -2x^2$ (narrow, opens down).

💡 Think of $|a|$ as the "squeeze factor" — bigger squeeze means narrower!

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2. The effect of coefficient 'a' on the parabola's direction and width (stretching)

The effect of coefficient 'a' on the parabola's direction and width (stretching)

The coefficient $a$ in $y = ax^2 + bx + c$ controls two critical properties: the direction (opening upward or downward) and the width (how narrow or wide the parabola appears). The sign of $a$ determines concavity, while its absolute value controls vertical stretching.

Intuition: Think of $a$ as a "shape controller"—its sign flips the parabola, and its size squeezes or spreads it vertically.

Core Rules:

If $a > 0$ : Parabola opens upward (U-shape)
If $a < 0$ : Parabola opens downward (inverted U-shape)
If $|a| > 1$ : Parabola is narrower (vertical stretch)
If $0 < |a| < 1$ : Parabola is wider (vertical compression)

Consequence: Larger absolute values of $a$ make the parabola steeper, while values closer to zero make it flatter. The vertex location remains unaffected by $a$ alone.

Example: $y = 3x^2$ is narrower and opens upward, while $y = -0.5x^2$ is wider and opens downward.

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Which of the following quadratic equations represents a parabola that opens downward?

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✖️ 3. Converting between standard form and vertex form

🔀 Switching Between Two Forms

Standard form is $y = ax^2 + bx + c$ .
Vertex form is $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex.
To go from standard to vertex: complete the square or use $h = -\frac{b}{2a}$ and find $k$ .
To go from vertex to standard: expand $(x - h)^2$ and simplify.
Both forms have the same $a$ value (the leading coefficient stays unchanged).

Convert $y = x^2 + 6x + 5$ : vertex is $(-3, -4)$ , so vertex form is $y = (x + 3)^2 - 4$ .

💡 Vertex form instantly shows you the turning point; standard form shows the y-intercept!

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3. Converting between standard form and vertex form

Converting between standard form and vertex form

A quadratic function can be expressed in standard form $y = ax^2 + bx + c$ or vertex form $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex. Converting between forms reveals different structural information about the parabola.

Intuition: Standard form shows intercepts easily; vertex form immediately displays the vertex location and makes transformations transparent.

Core Rules:

Standard to vertex: Complete the square on $ax^2 + bx$ to obtain $a(x - h)^2 + k$
Vertex to standard: Expand $a(x - h)^2 + k$ by distributing and combining like terms
Vertex coordinates: $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$ from standard form
Both forms have the same coefficient $a$

Consequence: Each form serves different purposes—vertex form for graphing transformations, standard form for finding intercepts algebraically.

Example: $y = 2x^2 - 8x + 5$ converts to $y = 2(x - 2)^2 - 3$ by completing the square, revealing vertex $(2, -3)$ .

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Convert the vertex form equation $y = 3(x - 2)^2 + 4$ into standard form.

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✖️ 4. Finding the maximum or minimum value of a quadratic function

📊 Locating the Extreme Value

If $a > 0$ , the parabola has a minimum value at the vertex.
If $a < 0$ , the parabola has a maximum value at the vertex.
The extreme value is the y-coordinate of the vertex, not the x-coordinate.
Calculate it using $y = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c$ or use vertex form directly.
This value represents the lowest or highest output the function can produce.

For $y = -2x^2 + 8x - 3$ : vertex x is $2$ , so maximum value is $-2(2)^2 + 8(2) - 3 = 5$ .

💡 The vertex y-value is always the answer for max/min problems!

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4. Finding the maximum or minimum value of a quadratic function

Finding the maximum or minimum value of a quadratic function

Every quadratic function $y = ax^2 + bx + c$ has either a minimum value (if $a > 0$ ) or a maximum value (if $a < 0$ ) occurring at the vertex. This extreme value is the y-coordinate of the vertex.

Intuition: Since parabolas have a single turning point, the vertex represents the absolute highest or lowest output value the function can produce.

Core Rules:

If $a > 0$ : The function has a minimum value at the vertex
If $a < 0$ : The function has a maximum value at the vertex
Extreme value: $y_{\text{extreme}} = f\left(-\frac{b}{2a}\right) = c - \frac{b^2}{4a}$
In vertex form $y = a(x - h)^2 + k$ , the extreme value is simply $k$

Consequence: Optimization problems involving quadratic relationships reduce to finding the vertex's y-coordinate.

Example: For $y = -x^2 + 6x - 5$ , since $a = -1 < 0$ , the maximum occurs at $x = 3$ , giving maximum value $y = 4$ .

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What is the minimum value of the quadratic function $y = 3(x - 4)^2 + 7$ ?

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✖️ 5. Applications: Projectile motion paths in physics and maximizing profit curves in business

🚀 Real-World Parabolas

Projectile motion: height versus time follows $h = -at^2 + vt + h_0$ where $a$ relates to gravity.
The maximum height occurs at the vertex of this parabola.
Profit curves: profit $P = -ax^2 + bx - c$ models how profit changes with production quantity.
Maximum profit happens at the vertex x-value (optimal production level).
Both applications use the vertex to find the best outcome (highest point or most profit).

A ball thrown follows $h = -5t^2 + 20t + 2$ ; max height is at $t = 2$ seconds, giving $h = 22$ meters.

💡 Vertex = peak performance in physics and business!

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5. Applications: Projectile motion paths in physics and maximizing profit curves in business

Applications: Projectile motion paths in physics and maximizing profit curves in business

Quadratic functions model real-world phenomena where relationships involve squared terms. In projectile motion, the height $h(t) = -gt^2 + v_0t + h_0$ traces a parabolic path under gravity. In business, profit functions $P(x) = -ax^2 + bx - c$ often exhibit parabolic behavior due to diminishing returns.

Intuition: Parabolas naturally describe situations with acceleration (physics) or optimal points balancing competing factors (economics).

Core Rules:

Projectile motion: Maximum height occurs at the vertex; $a < 0$ represents downward gravity
Business profit: Maximum profit occurs at the vertex when costs increase quadratically
Domain restrictions: Physical contexts often limit $x$ to non-negative values
The vertex provides the optimal solution (maximum height, maximum profit)

Consequence: Identifying the vertex directly solves optimization problems in these applied contexts.

Example: A ball's height $h(t) = -5t^2 + 20t + 2$ reaches maximum height 22 meters at $t = 2$ seconds.

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A toy rocket is launched into the air. Its height in meters after $t$ seconds is modeled by the function $h(t) = -5t^2 + 40t + 5$ .

Find the time $t$ in seconds when the rocket reaches its maximum height.

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Quadratic function and parabola ( $y = ax^2 + bx + c$ )

✖️ 1. Identifying vertex, axis of symmetry, and intercepts of a parabola

🎯 Finding Key Points on a Parabola

1. Identifying vertex, axis of symmetry, and intercepts of a parabola

Identifying vertex, axis of symmetry, and intercepts of a parabola

✖️ 2. The effect of coefficient 'a' on the parabola's direction and width (stretching)

🔄 How 'a' Controls the Parabola Shape

2. The effect of coefficient 'a' on the parabola's direction and width (stretching)

The effect of coefficient 'a' on the parabola's direction and width (stretching)

✖️ 3. Converting between standard form and vertex form

🔀 Switching Between Two Forms

3. Converting between standard form and vertex form

Converting between standard form and vertex form

✖️ 4. Finding the maximum or minimum value of a quadratic function

📊 Locating the Extreme Value

4. Finding the maximum or minimum value of a quadratic function

Finding the maximum or minimum value of a quadratic function

✖️ 5. Applications: Projectile motion paths in physics and maximizing profit curves in business

🚀 Real-World Parabolas

5. Applications: Projectile motion paths in physics and maximizing profit curves in business

Applications: Projectile motion paths in physics and maximizing profit curves in business

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