Quadratic Functions: Standard, Vertex, and Graphing

Key Concepts

A quadratic function is a function whose graph is a parabola. It is written in function notation as f(x), and by evaluating inputs we get outputs (f(x) is the same as y in a graph).
A function assigns each input x a unique output y; in a parabola this means no x-value corresponds to two different y-values on the graph.
Common forms of a quadratic equation and what they tell you about the graph:
- Standard form: $f(x)=ax^2+bx+c$ with descending powers of x. Here:
- The coefficient a determines the direction of the parabola (concave up if a>0, concave down if a<0).
- The constant term c is the y-intercept (the point where the graph crosses the y-axis) since f(0)=c.
- The b term affects the slope of the tangent near the y-intercept and the vertex location.
- Vertex form: $f(x)=a(x-h)^2+k$ where the vertex is at $(-h, k)$ . This form makes the vertex explicit and shows the axis of symmetry as the vertical line x = h.
- Factor form: $f(x)=a(x-r1)(x-r2)$ where the x-intercepts (zeros) are $x=r1$ and $x=r2$ when real.
A parabola is symmetric about its axis of symmetry, which is a vertical line passing through the vertex; for standard form, the axis is $x=-\frac{b}{2a}$ and the vertex is $\big(-\frac{b}{2a}, f(-\frac{b}{2a})\big)$ .
The discriminant $D=b^2-4ac$ from the quadratic formula determines the number and type of real roots (x-intercepts):
- If D>0, there are two real intercepts.
- If D=0, there is exactly one real intercept (tangent to the x-axis).
- If D<0, there are no real intercepts.
The quadratic formula gives the x-intercepts directly: $x=\frac{-b\,\pm\,\sqrt{b^2-4ac}}{2a}.$ The exact roots can be simplified, potentially producing surds like $-4\pm 3\sqrt{2}$ .
Intercepts and key points often guide graphing: evaluate a few points like f(0)=c, f(1), f(-1), etc., and use symmetry about the axis to find additional points.
Graphing tools:
- Desmos and graphing calculators can plot f(x) in function notation (Desmos uses f(x); calculators often use y= or function input).
- For calculators, turn off extra plots (Plot 1, Plot 2, Plot 3) if they are highlighted, so you only see the parabola you plotted.
- The trace (or cursor) feature lets you read coordinates of specific points on the graph (e.g., x-intercepts or a point like (1,7)).
Domain and range:
- For a quadratic, the domain is all real numbers. The range depends on the sign of a: concave up (a>0) gives a minimum value at the vertex; concave down (a<0) gives a maximum value at the vertex.
Conceptual analogies used in class:
- Like biology has characteristics and categories, a quadratic has attributes (a, b, c, vertex, intercepts) that describe its graph.

Standard Form: ax^2 + bx + c

Standard form: $f(x)=ax^2+bx+c$ with a≠0 to be a quadratic.
Descending order of exponents helps organize the work and identify the quadratic term, linear term, and constant term.
Key terms:
- Quadratic term: $ax^2$ (the coefficient is a)
- Linear term: $bx$ (the coefficient is b)
- Constant term: $c$
The y-intercept is the point where x=0, so the y-intercept is $c$ (the graph crosses the y-axis at (0,c)).
The value of a determines the direction of the parabola:
- If $a>0$ , the parabola opens upward (a
 abla), i.e., a smile; the graph is concave up.
- If $a<0$ , the parabola opens downward (a
 abla), i.e., a frown; the graph is concave down.
Vertex coordinates from standard form:
- $h = -\frac{b}{2a}$
- $k = f(h)$
- Vertex: $(h,k)$ and axis of symmetry: $x=h$
Example: For $f(x)=x^2+8x-2$ , we have $a=1, b=8, c=-2$ .
- Y-intercept: $f(0)=c=-2\;\Rightarrow\;(0,-2)$
- Hairline evaluation: $f(1)=1^2+8(1)-2=7\;\Rightarrow\;(1,7)$
- Vertex: $h=-\frac{b}{2a}=-\frac{8}{2(1)}=-4$
- $k=f(-4)=(-4)^2+8(-4)-2=16-32-2=-18$
- Vertex: $(-4,-18)$ ; axis of symmetry: $x=-4$
The graph’s shape is determined by a; here $a=1>0$ , so it opens upward (concave up), which is a smile.
X-intercepts (zeros) are the real solutions to $f(x)=0$ . A quadratic can have up to two real x-intercepts.
Using the standard form, we can also locate other points via symmetry: if you know one point (x1,y1), the reflection about the axis x=h gives another point (x2,y1) with x2 = 2h - x1.
Example symmetry for (1,7) about axis x=-4:
- Distance from axis to x=1 is $|1-(-4)|=5$
- Mirror point is at $x=-4-5=-9$ with the same y-value, so point is $(-9,7)$ .
Y-intercept and vertex together help sketch the parabola; intercepts help with calculating the exact crossing points on x-axis.

Vertex Form

Vertex form: $f(x)=a(x-h)^2+k$
Vertex is explicitly at $(-h, k)$ ; axis of symmetry is the vertical line $x=h$ .
In vertex form, a is the same as in standard form and controls stretch/boldness of the parabola.
Example: If the vertex is $(-4,-18)$ and $a=1$ , then
- Vertex form is $f(x)=(x+4)^2-18$
- Vertex: $(-4,-18)$ ; axis of symmetry: $x=-4$ ; concavity is determined by a (>0 opens up, <0 opens down).
To read the y-intercept from vertex form, evaluate at x=0:
- $f(0)=a(0-h)^2 + k = a h^2 + k$ where h is the horizontal shift (note: in the expression, h is the value inside the parentheses with a minus sign).
- For the example f(x) = (x+4)^2 - 18 with a=1, h=-4, k=-18, this gives $f(0)=(0+4)^2-18=16-18=-2$ , so y-intercept is (0,-2), consistent with the standard form.
Using vertex form for quick sketch:
- Plot the vertex (h,k).
- Draw the axis of symmetry x=h.
- Determine y-intercept by plugging x=0.
- If you want x-intercepts, you would solve f(x)=0 (either by factoring, completing the square, or using the quadratic formula).

Factor Form

Factor form (when real roots exist): $f(x)=a(x-r1)(x-r2)$ where the zeros are $x=r1$ and $x=r2$ .
This form makes x-intercepts immediately visible. If the quadratic can be factored over the reals, you can read off the roots directly.
Example insights:
- If the graph has two real x-intercepts, they are the r1 and r2 values.
- If it has a double root (D=0), the two intercepts coincide at the same x-value (the parabola is tangent to the x-axis).

The Quadratic Formula and Discriminant

Use the standard form set to zero: $ax^2+bx+c=0$
Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Discriminant: $D=b^2-4ac$ determines the number and type of real roots:
- If $D>0$ , two real x-intercepts.
- If $D=0$ , one real x-intercept (double root).
- If $D<0$ , no real x-intercepts (complex roots).
Example: For $f(x)=ax^2+bx+c$ with $a=1, b=8, c=-2$ :
- $D=b^2-4ac=64-4(1)(-2)=64+8=72$
- $x=\frac{-8\pm\sqrt{72}}{2}$
- Since $\sqrt{72}=6\sqrt{2}$ , this simplifies to
  $x=-4\pm 3\sqrt{2}$
- Approximate values: $-4-3\sqrt{2}\approx -8.2426$ and $-4+3\sqrt{2}\approx 0.2426$
Exact vs approximate: simplifying the radical gives the exact x-intercepts; decimals give approximate intercepts suitable for quick sketches.
Relation to graph: the two intercepts are the x-values where the graph crosses the x-axis (y=0).

Worked Example: f(x)=x^2+8x-2

Given function: $f(x)=x^2+8x-2$
Intercept checks:
- Y-intercept: $f(0)= -2\Rightarrow (0,-2)$
- A point on the graph: $f(1)=1+8-2=7\Rightarrow (1,7)$
Vertex calculation:
- $h=-\frac{b}{2a}=-\frac{8}{2(1)}=-4$
- $k=f(-4)=(-4)^2+8(-4)-2=16-32-2=-18$
- Vertex: $(-4,-18)$ ; axis of symmetry: $x=-4$
Using symmetry to find a second point from (1,7):
- Distance to axis: $|1-(-4)|=5$
- Mirror x-value: $-4-5=-9$ with same y, giving $(-9,7)$
X-intercepts (solve f(x)=0):
- Using quadratic formula with $a=1, b=8, c=-2$ :
  $D=b^2-4ac=64-4(1)(-2)=72$
  $x=\frac{-8\pm\sqrt{72}}{2}=\frac{-8\pm 6\sqrt{2}}{2}=-4\pm 3\sqrt{2}$
- Approximate intercepts: $x\approx -8.2426\quad\text{and}\quad x\approx 0.2426$
Exact intercepts: if desired, keep as $-4\pm 3\sqrt{2}$ .
Vertex form check:
- Since vertex is (-4,-18) and a=1, vertex form is $f(x)=(x+4)^2-18$ .
- This form confirms the vertex and axis of symmetry as above.
Summary of this example:
- Standard form yields y-intercept, vertex, axis of symmetry, and a pathway to x-intercepts via the discriminant or quadratic formula.
- Vertex form highlights the vertex directly and confirms the same axis and intercept behavior.

Graphing Tips and Tools

Desmos:
- You can enter the function using function notation, e.g., f(x) = x^2 + 8x - 2.
- If the graph isn’t fully in view, adjust the screen (zoom/pan) so the vertex and intercepts are visible.
- You can locate points like (1,7) or (-9,7) by placing the cursor on the graph; Desmos shows coordinates.
- You can compare Desmos graphs with calculator traces to ensure consistency.
Calculator (graphing mode):
- Enter y = x^2 + 8x - 2; make sure other plots are turned off (Plot 1, Plot 2, Plot 3 should not be highlighted).
- Use Trace to move along the curve and read coordinates (e.g., Trace at x=1 gives y=7, Trace at x=0 gives y=-2).
- For a quick symmetry check, compute a point (x1,y1) and reflect across the axis x=h to verify the matching point (2h - x1, y1).
Graph interpretation reminders:
- The graph’s shape (upward or downward) is dictated by the sign of a.
- The x-intercepts are the roots where f(x)=0; their existence and number depend on the discriminant.
- The vertex is the minimum or maximum value of the function depending on concavity.

Putting It All Together: Quick Reference

Given a quadratic in standard form, identify: $a,b,c$ from $f(x)=ax^2+bx+c$ .
Y-intercept: $f(0)=c$ ; Vertex x-coordinate: $h=-\frac{b}{2a}$ ; Vertex y-coordinate: $k=f(h)$ ; Vertex: $(h,k)$ ; Axis of symmetry: $x=h$ .
If you want the x-intercepts exactly, use the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ ; discriminant $D=b^2-4ac$ decides how many real intercepts you get.
If you prefer vertex form, rewrite as $f(x)=a(x-h)^2+k$ ; vertex is $(-h,k)$ and axis is $x=h$ .
If factorable, rewrite as $f(x)=a(x-r1)(x-r2)$ ; x-intercepts are $x=r1,r2$ .
For graphing practice, plot several known points (including intercepts, vertex, and a symmetric partner) to sketch accurately, then use a tool like Desmos or a calculator to verify.
Domain of a quadratic function is all real numbers; range depends on the sign of a and the vertex value.

Practice Checks (conceptual prompts)

If a>0, what can you say about the graph’s shape and minimum/maximum? (Concave up; has a minimum at the vertex.)
How would you determine the axis of symmetry from standard form? (Use $x=-\frac{b}{2a}$ .)
How can you quickly read off the vertex from vertex form? (Vertex is (h,k) in $f(x)=a(x-h)^2+k$ .)
What does the discriminant tell you about the x-intercepts without plotting? (Number and type of real roots.)
How can you verify a found x-intercept on Desmos or a calculator? (Use trace or cursor to read the coordinates and substitute back into f(x) to check near zero for approximate roots.)

// End of notes: these sections and formulas provide the key pieces needed to analyze, graph, and understand quadratic functions in standard, vertex, and factor forms.