Quadratic Functions, Inequalities, and Transformations of Graphs

Quadratic Functions and Their Graphs

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree 2, typically written as $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of a quadratic function is a parabola.

• Vertex: The highest or lowest point of the parabola, depending on whether it opens upward ($a > 0$) or downward ($a < 0$).

• Axis of Symmetry: A vertical line passing through the vertex, given by $x = -\frac{b}{2a}$.

• Zeroes (Roots): The x-values where $f(x) = 0$; these are the points where the graph crosses the x-axis.

• Y-intercept: The point where the graph crosses the y-axis, found by evaluating $f(0)$.

Example: Analyzing $f(x) = x^2 - 4$

• Zeroes: Set $x^2 - 4 = 0$ to find $x = -2$ and $x = 2$.

• Y-intercept: $f(0) = 0^2 - 4 = -4$.

• Vertex: Since $a = 1$, $b = 0$, the vertex is at $x = 0$, so $(0, -4)$.

• Intervals where $f(x) < 0$: The function is negative between the zeroes, i.e., for $-2 < x < 2$.

• Intervals where $f(x) \geq 0$: The function is non-negative for $x \leq -2$ or $x \geq 2$.

Factoring to Solve Quadratic Inequalities

To solve $x^2 - 4 < 0$, factor as $(x + 2)(x - 2) < 0$. The solution is $-2 < x < 2$.

To solve $x^2 - 4 \geq 0$, factor as $(x + 2)(x - 2) \geq 0$. The solution is $x \leq -2$ or $x \geq 2$.

Example: Solving $2x^2 - 3x - 2 = 0$

• Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• Here, $a = 2$, $b = -3$, $c = -2$.

• Calculate the discriminant: $(-3)^2 - 4(2)(-2) = 9 + 16 = 25$.

• Find the roots: $x = \frac{3 \pm 5}{4}$, so $x = 2$ and $x = -0.5$.

• Intervals where $2x^2 - 3x - 2 < 0$: The function is negative between the roots, i.e., for $-0.5 < x < 2$.

• Intervals where $2x^2 - 3x - 2 \geq 0$: The function is non-negative for $x \leq -0.5$ or $x \geq 2$.

Key Questions for Quadratic Graphs

• a.) What are the zeroes?

• b.) What is the y-intercept?

• c.) Where is the vertex?

• d.) When is the function less than or equal to zero?

• e.) When is the function greater than or equal to zero?

Application Example: Projectile Motion

The height of a golf ball after $t$ seconds is given by $s(t) = -16t^2 + 64t + 4$ (in feet). To find when the ball is at least 52 ft above the ground, solve $-16t^2 + 64t + 4 \geq 52$.

• Rearrange: $-16t^2 + 64t + 4 - 52 \geq 0 \implies -16t^2 + 64t - 48 \geq 0$.

• Solve for $t$ using the quadratic formula.

• From the graph, the ball is at 52 ft at $t = 1$ and $t = 3$ seconds. Between $t = 1$ and $t = 3$, the ball is higher than 52 ft.

Application Example: Area of a Circle

If a circle has an area of at least $225\pi$, what are the possible diameters?

• Area formula: $A = \pi r^2 \geq 225\pi$.

• Divide both sides by $\pi$: $r^2 \geq 225$.

• Take the square root: $r \geq 15$.

• Diameter $d = 2r \geq 30$.

Transformations of Graphs

Vertical and Horizontal Shifts

Transformations change the position or shape of a graph. The most common transformations are vertical and horizontal shifts.

• Vertical Shift: Adding or subtracting a constant $c$ to a function $f(x)$ results in $f(x) + c$ (up) or $f(x) - c$ (down).

• Horizontal Shift: Adding or subtracting a constant inside the function, $f(x + c)$ (left) or $f(x - c)$ (right).

Example: For $f(x) = |x|$, $f(x) + 2$ shifts the graph up 2 units, $f(x) - 2$ shifts it down 2 units. $f(x + 2)$ shifts left 2 units, $f(x - 2)$ shifts right 2 units.

Reflections

• Multiplying the function by $-1$ reflects the graph across the x-axis: $y = -f(x)$.

• Multiplying the input by $-1$ reflects the graph across the y-axis: $y = f(-x)$.

Stretching and Shrinking

• Multiplying the function by a constant $c$ ($c > 1$) stretches the graph vertically: $y = c f(x)$.

• If $0 < c < 1$, the graph is vertically shrunk.

• Multiplying the input by $c$ ($y = f(cx)$) compresses the graph horizontally if $c > 1$, and stretches it if $0 < c < 1$.

Example: $f(x) = |3x|$ is a horizontal compression by a factor of $1/3$; $f(x) = |x/3|$ is a horizontal stretch by a factor of 3.

Combined Transformations

Multiple transformations can be applied to a function. For example, $f(x) = -\sqrt{x - 2} + 2$ involves a reflection, a horizontal shift, and a vertical shift.

• Example: The graph of $f(x) = \sqrt{x}$ is shifted left 2 units and up 2 units, and reflected across the x-axis to get $f(x) = -\sqrt{x + 2} + 2$.

Summary Table: Common Graph Transformations

Transformation	Equation	Effect
Vertical Shift Up	$f(x) + c$	Shifts graph up by $c$ units
Vertical Shift Down	$f(x) - c$	Shifts graph down by $c$ units
Horizontal Shift Left	$f(x + c)$	Shifts graph left by $c$ units
Horizontal Shift Right	$f(x - c)$	Shifts graph right by $c$ units
Vertical Stretch	$c f(x)$, $c > 1$	Stretches graph vertically by $c$
Vertical Shrink	$c f(x)$, $0 < c < 1$	Shrinks graph vertically by $c$
Reflection over x-axis	$-f(x)$	Reflects graph over x-axis
Reflection over y-axis	$f(-x)$	Reflects graph over y-axis

Additional info:

• Understanding transformations is essential for graphing and analyzing functions in algebra and calculus.

• These concepts are foundational for solving equations, modeling real-world problems, and preparing for advanced mathematics.