Quadratic graphs are the heart of many SAT Math questions. They're those U-shaped curves you've probably seen before, and they're super important for modeling real-world situations like projectile motion or profit calculations.

Understanding the key parts of quadratic graphs - like the vertex, axis of symmetry, and intercepts - is crucial. These elements help you quickly sketch graphs and solve problems. Plus, knowing how to interpret these features in context can give you a big edge on the test.

Characteristics of Quadratic Functions

more resources to help you study

practice questions

Definition and Standard Form

• Quadratic function represents polynomial function of degree 2
• Standard form expressed as $f(x) = ax^2 + bx + c$, where a, b, and c are constants and $a ≠ 0$
• Graph forms smooth, U-shaped curve called parabola
• Parabola exhibits symmetry about vertical line (axis of symmetry)

Key Points on the Graph

• Y-intercept occurs where graph intersects y-axis, found by evaluating $f(0)$
• X-intercepts (roots) represent points where graph intersects x-axis
• Solve equation $ax^2 + bx + c = 0$ to find x-intercepts
• Vertex marks point where graph reaches maximum or minimum value
• Vertex always lies on axis of symmetry

Axis of Symmetry, Vertex, and Opening

Axis of Symmetry

• Vertical line divides parabola into two equal halves
• Equation for axis of symmetry $x = -b/(2a)$
• Coefficients a and b derived from standard form of quadratic function

Vertex Formula and Significance

• Vertex calculated using formula $(-b/(2a), f(-b/(2a)))$
• Represents highest or lowest point on parabola
• Critical for determining maximum or minimum values of function

Direction of Opening

• Determined by sign of leading coefficient a in standard form
• Parabola opens upward when $a > 0$, indicating minimum value
• Parabola opens downward when $a < 0$, indicating maximum value

Maximum and Minimum Values

Calculating Extrema

• Determine vertex using formula $(-b/(2a), f(-b/(2a)))$
• Y-coordinate of vertex represents maximum value for downward-opening parabola $(a < 0)$
• Y-coordinate of vertex represents minimum value for upward-opening parabola $(a > 0)$

Real-World Applications

• Maximum value may represent peak height (projectile motion)
• Minimum value could indicate lowest production cost (manufacturing)
• Extrema often crucial in optimization problems (maximizing profit, minimizing expenses)

Graphing Quadratic Functions

Manual Graphing Process

• Identify key characteristics: y-intercept, x-intercepts, vertex, opening direction
• Plot y-intercept and x-intercepts (if any) on coordinate plane
• Calculate and plot vertex using $(-b/(2a), f(-b/(2a)))$
• Connect points with smooth U-shaped curve, considering opening direction

Transformations from Parent Function

• Parent function: $f(x) = x^2$
• Vertical shifts: $f(x) = x^2 + k$ (up k units for $k > 0$, down |k| units for $k < 0$)
• Horizontal shifts: $f(x) = (x - h)^2$ (right h units for $h > 0$, left |h| units for $h < 0$)
• Vertical stretches/compressions: $f(x) = a(x^2)$ (stretch by |a| for $|a| > 1$, compress for $0 < |a| < 1$)
• Reflections: $f(x) = -x^2$ (reflects graph across x-axis)

Interpreting Quadratic Graphs

Real-World Contexts

• Model various situations (projectile motion, business profit, rectangular area)
• X-axis typically represents independent variable (time, quantity)
• Y-axis usually depicts dependent variable (height, profit, area)

Interpreting Key Features

• Vertex signifies maximum or minimum point in context (highest point, lowest cost)
• X-intercepts may indicate process start/end points or financial break-even points
• Y-intercept often represents initial value when independent variable equals zero
• Shape and opening direction provide insights into rate of change and overall behavior of modeled phenomenon