Algebra II CPE: Unit 2 Test - Perry (Ross)

Quadratic Function

A function that can be written in the form f(x) = a(x − h)² + k, where a ≠ 0.

Parabola

The U-shaped graph of a quadratic function.

Parent Function

f(x) = x² is the most basic quadratic function.

Vertex Form

f(x) = a(x − h)² + k, where (h, k) is the vertex.

Vertex

The highest or lowest point on a parabola; given by (h, k) in vertex form.

Axis of Symmetry (AOS)

A vertical line that divides a parabola into two mirror images; its equation is x = h.

Standard Form

f(x) = ax² + bx + c.

Intercept Form

f(x) = a(x − p)(x − q).

Horizontal Translation Rule

f(x − h)² shifts the parabola right if h > 0, left if h < 0.

Vertical Translation Rule

f(x) + k shifts the parabola up if k > 0, down if k < 0.

Reflection in x-axis

Multiply the function by -1; f(x) becomes -f(x).

Reflection in y-axis

Replace x with -x; f(x) becomes f(-x).

Vertical Stretch

Occurs when |a| > 1; the parabola becomes narrower.

Vertical Shrink

Occurs when 0 < |a| < 1; the parabola becomes wider.

Horizontal Stretch

Occurs when 0 < a < 1 in f(ax)²; the parabola stretches away from the y-axis.

Horizontal Shrink

Occurs when a > 1 in f(ax)²; the parabola moves closer to the y-axis.

Combination Transformation Example

f(x) = -3x² + 4 represents reflection in x-axis, vertical stretch by 3, and translation 4 units up.

How to Identify Transformations

Compare new function g(x) to parent f(x) = x² to see if it’s shifted, stretched, or reflected.

a Value Meaning

Controls reflection and stretch/shrink.

h Value Meaning

Controls horizontal shift (left/right).

k Value Meaning

Controls vertical shift (up/down).

AOS from Vertex Form

x = h.

AOS from Standard Form

x = -b ÷ (2a).

Vertex from Standard Form

(-b ÷ 2a, f(-b ÷ 2a)).

AOS from Intercept Form

x = (p + q) ÷ 2.

x-intercepts (from Intercept Form)

p and q are the x-intercepts of the graph.

Direction of Opening

Parabola opens up if a > 0, down if a < 0.

Finding Vertex Example

For y = -x² + 2x - 5, x = -b ÷ 2a = -2 ÷ (2×-1) = 1, so vertex is (1, -4).

How to Graph in Vertex Form

1. Identify a, h, k. 2. Plot vertex (h, k). 3. Draw AOS x = h. 4. Choose points on one side, reflect across AOS. 5. Draw parabola.

How to Graph in Standard Form

1. Find vertex (x = -b ÷ 2a). 2. Find y-intercept (c). 3. Plot vertex, y-intercept, and reflected points. 4. Sketch parabola.

How to Graph in Intercept Form

1. Identify p, q (x-intercepts). 2. Find vertex using x = (p + q)/2. 3. Plug x in to find y. 4. Sketch parabola.

Standard Form to Vertex Form

Complete the square or use vertex formula.

Vertex Formula Reminder

x = -b ÷ (2a).

Finding Minimum Value

When a > 0, vertex gives the lowest y-value.

Finding Maximum Value

When a < 0, vertex gives the highest y-value.

Domain of Any Quadratic

All real numbers (-∞, ∞).

Range When a > 0

y ≥ k or [k, ∞).

Range When a < 0

y ≤ k or (-∞, k].

Increasing Interval (a > 0)

To the right of the vertex.

Decreasing Interval (a > 0)

To the left of the vertex.

Increasing Interval (a < 0)

To the left of the vertex.

Decreasing Interval (a < 0)

To the right of the vertex.

Example of Minimum Function

f(x) = 2x² + 4x + 1 has a > 0, so vertex gives minimum.

Example of Maximum Function

f(x) = -x² + 2x - 5 has a < 0, so vertex gives maximum.

Converting Vertex to Standard Form

Expand a(x − h)² + k to ax² − 2ahx + (ah² + k).

Identifying a, b, c

From standard form f(x) = ax² + bx + c.

y-intercept

The point (0, c) where the graph crosses the y-axis.

Shape of Graph When |a| > 1

Parabola is narrow.

Shape of Graph When |a| < 1

Parabola is wide.

If f(x) = (x + 2)² + 1

Shifted 2 left, 1 up.

If f(x) = -(4x)²

Reflection in x-axis and horizontal shrink by ¼.

If f(x) = 3x² - 2

Vertical stretch by 3 and 2 units down.

Formula Connection Between Forms

Vertex

Real-life Meaning of Vertex

Represents the highest or lowest point of a quadratic relationship, such as profit, height, or area.

Summary of Transformations

h = left/right shift; k = up/down shift

a

stretch/shrink/flip

Parent Parabola Equation

y = x²

Example of Reflection in y-axis

f(-x) = (-x)² = x², so reflection over y-axis doesn’t change y = x².

Shortcut for Vertex Location (Standard Form)

Plug -b ÷ (2a) into f(x) to find y.

Tip for Graphing Quickly

Vertex gives middle of symmetry; choose equal x-values on both sides.

Intercept Meaning

Points where the graph crosses the x-axis or y-axis.

Equation for Range When Opens Up

y ≥ f(-b ÷ 2a).

Equation for Range When Opens Down

y ≤ f(-b ÷ 2a).

AOS Reminder

Always passes through the vertex.

Example Full Analysis

For f(x) = -7x² + 42x + 11, vertex at (3, 74); opens down; maximum = 74; domain all reals; range y ≤ 74.