Quadratic Functions Review

Horizontal Transformations of Quadratic Functions

• Replacing a parent function $f(x)$ with $f(x + k)$ results in a horizontal shift (translation) of the graph by $k$ units.

• Effect of the Constant $k$ on the Graph:

  • If $k > 0$ (positive constant inside the function argument), the graph of $f(x)$ shifts horizontally to the left by $k$ units.

  • If $k < 0$ (negative constant inside the function argument), the graph of $f(x)$ shifts horizontally to the right by $|k|$ units.

• Coordinate Point Mapping:

  • Under the transformation $f(x) \rightarrow f(x + k)$, every point $(x, y)$ on the original graph maps to $(x - k, y)$.

  • The y-coordinates of all points remain completely unchanged, while the x-coordinates decrease by $k$ units.

• Impact on Key Features:

  • Axis of Symmetry: Shifts horizontally from $x = 0$ (for standard parent function $f(x) = x^2$) to $x = -k$.

  • Vertex: Shifts from $(0, 0)$ to $(-k, 0)$.

  • Range: Remains unchanged, as vertical position is unaffected.

  • Domain: Continuous across all real numbers, (-\text{∑}, \text{∑}).

Vertical Transformations of Quadratic Functions

• Replacing a function $f(x)$ with $f(x) + k$ results in a vertical shift (translation) of the graph by $k$ units.

• Effect of the Constant $k$ on the Graph:

  • If $k > 0$ (positive constant added outside the function), the graph shifts vertically upwards by $k$ units.

  • If $k < 0$ (negative constant subtracted outside the function), the graph shifts vertically downwards by $|k|$ units.

• Coordinate Point Mapping:

  • Under the transformation $f(x) \rightarrow f(x) + k$, every point $(x, y)$ on the original graph maps to $(x, y + k)$.

  • The x-coordinates of all points remain completely unchanged, while the y-coordinates increase by $k$ units.

• Impact on Key Features:

  • Axis of Symmetry: Remains unaffected at $x = 0$ (unless horizontal shifts are also present).

  • Vertex: Shifts from $(0, 0)$ to $(0, k)$.

  • Extreme Values: The minimum or maximum value of the function changes directly by adding $k$ (new extreme value is $k$).

  • Range: Changes from [0, \text{∑}) to [k, \text{∑}) for upward-opening parabolas, or from (-\text{∑}, 0] to (-\text{∑}, k] for downward-opening parabolas.

Vertical Dilations and Reflections

• Replacing a function $f(x)$ with $k f(x)$ scales or reflects the graph vertically by a multiplicative factor $k$

• Effect of the Multiplier $k$ on the Shape of the Graph:

  • Vertical Stretch: If $|k| > 1$, the graph is stretched vertically by a factor of $|k|$, making the parabola appear narrower.

  • Vertical Compression: If $0 < |k| < 1$, the graph is compressed vertically by a factor of $|k|$, making the parabola appear wider.

  • Axis Reflection: If $k < 0$, the graph is reflected across the x-axis, changing its opening direction.

• Coordinate Point Mapping:

  • Under the transformation $f(x) \rightarrow k f(x)$, every point $(x, y)$ on the original graph maps to $(x, k y)$.

  • The x-coordinates remain identical while all y-coordinates are multiplied by $k$

• Combination of Stretch and Reflection:

  • If $k = -2$, the graph reflects across the x-axis and undergoes a vertical stretch by a factor of $2$

  • If $k = -\frac{1}{2}$, the graph reflects across the x-axis and undergoes a vertical compression by a factor of $\frac{1}{2}$

Equivalent Forms of Quadratic Functions

• Quadratic functions can be represented in three primary algebraic forms, each revealing distinct geometric properties of the parabola.

• Standard Form:

  • Expression: $f(x) = a x^2 + b x + c$, where $a \neq 0$

  • Revealed Properties: Directly shows the y-intercept at $(0, c)$ and whether the parabola opens upward ($a > 0$) or downward ($a < 0$).

• Vertex Form:

  • Expression: $f(x) = a(x - h)^2 + k$, where $a \neq 0$

  • Revealed Properties: Directly identifies the vertex $(h, k)$, the axis of symmetry $x = h$, and the vertical dilation/orientation factor $a$

• Factored Form (Intercept Form):

  • Expression: $f(x) = a(x - r_1)(x - r_2)$, where $a \neq 0$

  • Revealed Properties: Directly identifies the zeros (x-intercepts) at $(r_1, 0)$ and $(r_2, 0)$.

• Converting Between Equivalent Forms:

  • Standard to Vertex: Completed by completing the square or calculating $h = \frac{-b}{2a}$ and $k = f(h)$.

  • Vertex to Standard: Expanded algebraically via $(x - h)^2 = x^2 - 2 h x + h^2$ and simplified.

  • Factored to Standard: Expanded using polynomial distribution (FOIL) and multiplying by scalar $a$

Factoring Quadratic Expressions with Leading Coefficient a = 1

• Structure of Quadratic Expression:

  • Form: $x^2 + b x + c$

  • Target Factored Form: $(x + p)(x + q)$

• Step-by-Step Factoring Procedure:

  • Step 1: Identify coefficients $b$ (linear coefficient) and $c$ (constant term).

  • Step 2: Find two real numbers $p$ and $q$ that satisfy two simultaneous requirements:

  • Product Rule: $p \times q = c$

  • Sum Rule: $p + q = b$

  • Step 3: Rewrite the expression as the product of two binomials: $(x + p)(x + q)$.

• Sign Rules for Factoring:

  • If $c > 0$ and $b > 0$: Both $p$ and $q$ must be positive.

  • If $c > 0$ and $b < 0$: Both $p$ and $q$ must be negative.

  • If $c < 0$: One factor is positive and one factor is negative; the sign of the larger absolute value matches the sign of $b$

Identifying Zeros, Extreme Values, and Symmetry via Factoring

• Determining Zeros (x-intercepts):

  • Set the factored function equal to zero: $a(x - r_1)(x - r_2) = 0$

  • Apply the Zero Product Property: $x - r_1 = 0$ or $x - r_2 = 0$

  • Real Zeros: $x = r_1$ and $x = r_2$. Points on graph are $(r_1, 0)$ and $(r_2, 0)$.

• Determining Axis of Symmetry:

  • Due to parabolic symmetry, the axis of symmetry is located precisely halfway between the two zeros.

  • Formula for Axis of Symmetry: $x = \frac{r_1 + r_2}{2}$

• Determining Extreme Values (Vertex):

  • The x-coordinate of the vertex $h$ is equal to the axis of symmetry value: $h = \frac{r_1 + r_2}{2}$.

  • The y-coordinate of the vertex $k$ is obtained by substituting $h$ into the function: k = f\right(\frac{r_1 + r_2}{2}\right).

  • Type of Extremum:

  • If $a > 0$, the parabola opens upward, providing a absolute minimum value of $k$

  • If $a < 0$, the parabola opens downward, providing an absolute maximum value of $k$

Graphing Quadratic Functions from Vertex Form

• Equation Structure: $f(x) = a(x - h)^2 + k$

• Identification of Key Features:

  • Vertex: Read directly from equation parameters as $(h, k)$. Note that the sign inside the parenthesis is negated.

  • Axis of Symmetry: The vertical line defined by equation $x = h$

  • Direction of Opening:

  • Opens upward if $a > 0$

  • Opens downward if $a < 0$

  • Y-Intercept: Found by substituting $x = 0$ into the equation: $f(0) = a(0 - h)^2 + k = a h^2 + k$

• Step-by-Step Procedure for Graphing:

  • Step 1: Plot the vertex $(h, k)$ on the coordinate plane.

  • Step 2: Draw the vertical axis of symmetry line $x = h$ as a dashed line.

  • Step 3: Determine the direction of opening and vertical stretch factor $a$

  • Step 4: Calculate and plot additional reference points using step pattern relative to vertex $(h \text{±} 1, k + a)$, $(h \text{±} 2, k + 4a)$, or by finding the y-intercept.

  • Step 5: Reflect points across the axis of symmetry $x = h$ to ensure symmetric accuracy.

  • Step 6: Connect points with a smooth, continuous U-shaped curve with arrows at both ends.

Writing Equations of Parabolas Given Specific Attributes

• Attributes Provided:

  • Vertex $(h, k)$

  • Axis of symmetry $x = h$

  • Direction of opening (upward or downward)

  • Additional point on parabola $(x_1, y_1)$

• Procedure to Derive Equation in Vertex Form:

  • Step 1: Substitute the known vertex coordinates $(h, k)$ into vertex form equation: $f(x) = a(x - h)^2 + k$

  • Step 2: Substitute the coordinates of the given additional point $(x_1, y_1)$ into equation for $x$ and $f(x)$: $y_1 = a(x_1 - h)^2 + k$

  • Step 3: Solve algebraically for parameter $a$:

    $a = \frac{y_1 - k}{(x_1 - h)^2}$

• Step 4: Verify that the sign of $a$ matches the specified direction of opening:

  • Upward opening requires $a > 0$

  • Downward opening requires $a < 0$

• Step 5: Write final equation substituting numerical values for $a$, $h$, and $k$

Questions and Discussion

• Review Questions & Diagnostic Focus:

  • Identifying transformation types from equation modifications ($f(x + k)$, $f(x) + k$, and $k f(x)$).

  • Factoring quadratics with $a = 1$ efficiently and determining x-intercepts.

  • Converting between quadratic forms to state features such as vertex, zeros, axis of symmetry, and minimum/maximum values.

Here are a few practice problems based on your notes to test your understanding:

1. Horizontal and Vertical Transformations
   Describe the transformations required to convert the parent function $f(x) = x^2$ into $g(x) = (x + 3)^2 - 7$. State the vertex and axis of symmetry.

2. Factoring Quadratic Expressions
   Factor the quadratic expression $x^2 - 8x + 15$. Use your factored form to determine the zeros of the function $f(x) = x^2 - 8x + 15$.

3. Converting to Vertex Form
   Rewrite $f(x) = x^2 + 6x + 5$ in vertex form $f(x) = a(x - h)^2 + k$ by completing the square or finding the vertex. Identify the extreme value and state whether it is a minimum or maximum.

4. Writing the Equation of a Parabola
   A parabola has a vertex at $(3, -2)$ and passes through the point $(5, 6)$. Write its equation in vertex form.

Try solving these, and let me know if you would like to check your answers or see step-by-step solutions!

Here are 5 practice problems based on your notes: 1. Horizontal and Vertical Transformations: Describe the transformations required to convert the parent function $f(x) = x^2$ into $g(x) = -2(x - 4)^2 + 5$. State the vertex and axis of symmetry. 2. Factoring and Zeros: Factor the quadratic expression $x^2 - 7x + 12$ and find the zeros of $f(x) = x^2 - 7x + 12$. Use the zeros to determine the axis of symmetry and the vertex. 3. Writing Equations in Vertex Form: A parabola has a vertex at $(-2, 3)$ and passes through the point $(1, -15)$. Write its equation in vertex form $f(x) = a(x - h)^2 + k$. 4. Converting Between Forms: Convert $f(x) = (x - 2)(x + 6)$ into standard form $f(x) = a x^2 + b x + c$ and identify its y-intercept. 5. Identifying Key Features: For the function $f(x) = (x + 1)^2 - 9$, find the y-intercept, x-intercepts, vertex, minimum value, and range.