Graphing Quadratic Functions

📈 Graphing Quadratic Functions

Part 1 of 5 — The Parabola & Standard Form

Topics in This Part

🔑 Key Concept: Every quadratic function $y = ax^2 + bx + c$ graphs as a parabola — a smooth, symmetric U-shaped curve. Before we plot anything, we'll learn to read the equation and predict what the graph looks like.

What Is a Quadratic Function?

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

$y = ax^2 + bx + c \qquad (a \ne 0)$

The three numbers $a$, $b$, and $c$ are the coefficients:

⚠️ The defining feature is the $x^2$ term. If $a = 0$, there is no $x^2$, and the equation is just a line — not a quadratic.

Examples that ARE quadratics: $y = x^2$, $\;y = 3x^2 - 2x + 1$, $\;y = -x^2 + 4$. NOT quadratics: $y = 2x + 5$ (no $x^2$), $\;y = x^3 - x$ (highest power is $3$).

Concept Check 🎯

Which Way Does It Open?

The sign of the leading coefficient $a$ alone tells you the parabola's direction:

💡 Memory hook: A positive $a$ makes a happy (smiling) parabola that opens up. A negative $a$ makes a frowning parabola that opens down.

Examples:

• $y = x^2 - 6x + 5$ has $a = 1 > 0$, so it opens up (has a minimum).
• $y = -2x^2 + 8x - 6$ has $a = -2 < 0$, so it opens down (has a maximum).

Width and the $y$-intercept

Width is also set by $a$ — specifically by how large $|a|$ is:

• $|a| > 1$ → narrower (steeper) than $y = x^2$. Example: $y = 4x^2$.
• $0 < |a| < 1$ → wider (flatter) than $y = x^2$. Example: $y = \tfrac{1}{2}x^2$.

The $y$-intercept is the easiest point to find: set $x = 0$.

$y = a(0)^2 + b(0) + c = c$

So the parabola always crosses the $y$-axis at $(0, c)$ — the constant term, read straight off the equation.

🔑 Read it instantly: For $y = ax^2+bx+c$, the $y$-intercept is $(0, c)$. For $y = x^2 - 6x + 5$, it is $(0, 5)$.

Read the Equation 🔽

For each quadratic, decide its features directly from standard form.

Read Off the Coefficients 🧮

For $y = -2x^2 + 6x - 9$, enter each value.

1) The leading coefficient $a = \,?$ 2) The constant term $c = \,?$ 3) The $y$-coordinate of the $y$-intercept $= \,?$

📈 Graphing Quadratic Functions

Part 2 of 5 — Vertex & Axis of Symmetry

🔑 The Idea: The single most important point on a parabola is its vertex — the turning point. From standard form you can find it with one formula, and from there the rest of the graph falls into place.

The Axis of Symmetry

Every parabola is symmetric about a vertical line through its vertex, called the axis of symmetry. For $y = ax^2 + bx + c$, that line is:

$x = -\frac{b}{2a}$

This $x$-value is also the $x$-coordinate of the vertex.

Worked Example: $y = x^2 - 6x + 5$

Here $a = 1$ and $b = -6$:

$x = -\frac{b}{2a} = -\frac{-6}{2(1)} = \frac{6}{2} = 3$

So the axis of symmetry is $x = 3$.

⚠️ Watch the signs. The formula has a negative out front. With $b = -6$, you get $-(-6) = +6$ on top — a positive result.

Concept Check 🎯

Finding the Whole Vertex

The vertex is the point $\left(x_v,\ y_v\right)$ where:

1. $x_v = -\dfrac{b}{2a}$ (axis of symmetry)
2. $y_v$ = the function evaluated at $x_v$ — substitute $x_v$ back in.

Worked Example: $y = x^2 - 6x + 5$

We found $x_v = 3$. Now plug $x = 3$ into the original:

$y_v = (3)^2 - 6(3) + 5 = 9 - 18 + 5 = -4$

So the vertex is $(3, -4)$. Since $a = 1 > 0$, the parabola opens up, and $-4$ is the minimum value.

Worked Example: $y = -x^2 + 4x + 1$

Step 1 — axis of symmetry ($a = -1,\ b = 4$):

$x_v = -\frac{4}{2(-1)} = -\frac{4}{-2} = 2$

Step 2 — substitute $x = 2$:

$y_v = -(2)^2 + 4(2) + 1 = -4 + 8 + 1 = 5$

Vertex $(2, 5)$. Because $a = -1 < 0$, the parabola opens down, so $5$ is the maximum value.

💡 Once you have the vertex, symmetry gives you free points: any point $d$ units to the right of the axis has a twin $d$ units to the left at the same height.

Build the Vertex 🔽

Work through $y = 2x^2 - 8x + 3$ one step at a time.

Find the Vertex 🧮

For $y = x^2 + 2x - 8$, find the vertex.

1) Axis of symmetry: $x = \,?$ 2) Vertex $y$-coordinate $= \,?$ 3) Is the vertex a min or a max? (type min or max*)*

📈 Graphing Quadratic Functions

Part 3 of 5 — Intercepts, Roots & Factored Form

🔑 Why it matters: The points where a parabola crosses the $x$-axis are its $x$-intercepts (also called roots or zeros). They anchor the graph and tell you where $y = 0$.

Finding $x$-intercepts

An $x$-intercept is where the graph touches the $x$-axis, so $y = 0$. To find them, solve:

$ax^2 + bx + c = 0$

When the quadratic factors, this is quick.

Worked Example: $y = x^2 - 2x - 3$

Set $y = 0$ and factor:

$x^2 - 2x - 3 = 0 \;\Rightarrow\; (x - 3)(x + 1) = 0$

By the zero-product rule, each factor can be $0$:

$x - 3 = 0 \;\Rightarrow\; x = 3 \qquad x + 1 = 0 \;\Rightarrow\; x = -1$

So the $x$-intercepts are $(3, 0)$ and $(-1, 0)$.

💡 Bonus symmetry trick: the axis of symmetry sits exactly halfway between the roots. Here that is $\dfrac{3 + (-1)}{2} = 1$, so $x = 1$ — matching $-\frac{b}{2a} = \frac{2}{2} = 1$. ✓

Concept Check 🎯

How Many $x$-intercepts? The Discriminant

A parabola can cross the $x$-axis twice, once, or never. The discriminant $\Delta = b^2 - 4ac$ tells you which, without solving:

Examples:

• $y = x^2 - 5x + 6$: $\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0$ → two intercepts.
• $y = x^2 - 4x + 4$: $\Delta = 16 - 16 = 0$ → one (a double root at $x = 2$).
• $y = x^2 + 2x + 5$: $\Delta = 4 - 20 = -16 < 0$ → none.

Count the Intercepts 🧮

Use $\Delta = b^2 - 4ac$. Enter the discriminant value, then the number of $x$-intercepts.

1) $y = x^2 - 6x + 9$: $\Delta = \,?$ 2) Same equation: number of $x$-intercepts $= \,?$ 3) $y = x^2 + x + 4$: number of $x$-intercepts $= \,?$

Factored Form Hands You the Roots

A quadratic written in factored form displays its roots directly:

$y = a(x - r_1)(x - r_2) \quad\Longrightarrow\quad \text{roots } x = r_1,\ x = r_2$

⚠️ Sign flip: $(x - 3)$ gives the root $x = +3$; $(x + 1) = (x - (-1))$ gives the root $x = -1$. The root is the opposite of the number inside the parentheses.

Example: $y = (x + 4)(x - 2)$

Roots: $x = -4$ and $x = 2$. The axis of symmetry is halfway between: $\dfrac{-4 + 2}{2} = -1$, so $x = -1$.

From Factored Form 🔽

Read features straight from $y = (x - 5)(x + 1)$.

📈 Graphing Quadratic Functions

Part 4 of 5 — Vertex Form & Transformations

🔑 Big Payoff: Vertex form $y = a(x - h)^2 + k$ shows the vertex $(h, k)$ at a glance and reveals exactly how the basic parabola $y = x^2$ has been shifted and stretched.

Vertex Form

$y = a(x - h)^2 + k \quad\Longrightarrow\quad \text{vertex } (h,\, k)$

⚠️ Sign trap: $(x - h)$ means the $x$-coordinate is $+h$. So $(x - 3)^2$ has $h = 3$, but $(x + 2)^2 = (x - (-2))^2$ has $h = -2$.

The value of $a$ works exactly as in standard form: its sign sets the opening direction, and its size sets the width.

Concept Check 🎯

Transformations of $y = x^2$

Read vertex form as a recipe for moving the parent parabola $y = x^2$ (vertex at the origin):

Example: $y = 2(x - 3)^2 + 1$

Start at $y = x^2$, then: shift right 3, up 1, and stretch by $2$ (narrower). The new vertex is $(3, 1)$.

💡 The horizontal shift uses the opposite sign of $h$ inside the parentheses; the vertical shift $k$ uses its sign as-is.

Describe the Transformation 🔽

Compare each graph to the parent $y = x^2$.

Vertex Form Practice 🧮

For $y = -3(x + 2)^2 + 7$, find:

1) The vertex $x$-coordinate $= \,?$ 2) The vertex $y$-coordinate $= \,?$ 3) Does it open up or down? (type up or down*)*

📈 Graphing Quadratic Functions

Part 5 of 5 — Putting It All Together & Mastery Check

You can now read a parabola's direction, width, vertex, axis of symmetry, and intercepts straight from an equation. Part 5 ties these into a full graphing routine and finishes with an Exit Quiz.

The Graphing Routine

To sketch any $y = ax^2 + bx + c$, gather five facts:

Plot the vertex, the intercepts, and use symmetry to mirror points across the axis — then connect with a smooth U.

🔑 One worked sketch: For $y = x^2 - 4x + 3$ → opens up; axis $x = 2$; vertex $(2, -1)$; $y$-intercept $(0, 3)$; $x$-intercepts $(1, 0)$ and $(3, 0)$.

Full Analysis 🔽

Analyze $y = x^2 - 10x + 21$ completely.

Mixed Practice 🧮

For $y = x^2 + 8x + 15$:

1) Axis of symmetry: $x = \,?$ 2) Vertex $y$-coordinate $= \,?$ 3) Smaller $x$-intercept: $x = \,?$ (the more negative root)

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.