2.2 Sketching Quadratic Graphs

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📖 Main Content

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Part 1: What is a Quadratic Function?

A quadratic function is any function that can be written in the form:

$f(x) = ax^2 + bx + c$

where:

• $a$, $b$, and $c$ are numbers (called constants)
• $a \neq 0$ (meaning $a$ cannot be zero — otherwise it wouldn't be quadratic)

When you draw the graph of a quadratic function, you always get a smooth, curved shape called a parabola. Think of it as a U-shape or an upside-down U-shape.

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Part 2: The Shape of the Parabola — Which Way Does It Open?

The value of $a$ (the number in front of $x^2$) tells you which direction the parabola opens.

Remember: If $a$ is positive, the curve looks like a valley (has a bottom). If $a$ is negative, the curve looks like a hill (has a top).

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Part 3: Key Features of a Parabola

Every parabola has these important features that you need to find before sketching:

① The Turning Point (also called the Vertex)

• This is the point where the graph changes direction
• If it's the lowest point → called the minimum point
• If it's the highest point → called the maximum point
• Also known as the stationary point
• Written as a coordinate pair: $(h, k)$

② The Line of Symmetry

• Every parabola is perfectly symmetrical
• The line of symmetry is a vertical line that passes through the turning point
• It divides the parabola into two identical mirror halves
• Its equation is always written as $x = h$, where $h$ is the x-coordinate of the turning point

③ The y-intercept

• This is where the curve crosses the y-axis (where $x = 0$)
• Find it by substituting $x = 0$ into the function

④ The x-intercepts (roots)

• These are where the curve crosses the x-axis (where $y = 0$)
• Find them by solving $f(x) = 0$
• A parabola can have 0, 1, or 2 x-intercepts