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MathChapter 11: Quadratic Functions

Notes

Graphs of Quadratic Equations Notes

Read and connect standard, vertex, and factored forms using exact vertices, symmetry axes, intercepts, and digitally generated parabolas.

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A quadratic function has standard form \(f(x)=ax^2+bx+c\), where \(a\ne0\). Its graph is a parabola. Algebra and graph features tell the same story: the leading coefficient controls opening, the vertex gives the extreme value, and real zeros become horizontal-axis intercepts.

Learning objectives

  • Identify opening direction, vertex, symmetry axis, and intercepts.
  • Use \(x=-\frac{b}{2a}\) and evaluate the function to find the vertex.
  • Recognize what standard, vertex, and factored forms reveal immediately.
  • Confirm that every graph label agrees with its exact equation.

Quadratic vocabulary

Parabola
The curved graph of a quadratic function. It opens upward when \(a>0\) and downward when \(a<0\).
Vertex and axis of symmetry
The vertex is the parabola's minimum or maximum point. The vertical line through it is the axis of symmetry, \(x=-\frac{b}{2a}\).
Roots, zeros, solutions, and horizontal-axis intercepts
For \(f(x)=0\), these terms describe the input values where the parabola meets the horizontal axis. A quadratic can have zero, one, or two real roots.

Opening direction and extreme value

The sign of the leading coefficient

Two exact parabolas compare a positive leading coefficient with a negative leading coefficient. Each graph labels its vertex, roots, and symmetry axis.

Positive leading coefficientUpward-opening parabola with vertex at 0, -4, symmetry axis at horizontal coordinate 0, 2 real horizontal-axis intercepts at -2 and 2, and vertical-axis intercept -4.-6-5-4-3-2-1123456-5-4-3-2-1123456xyvertex (0,-4)root (-2,0)root (2,0)axis of symmetry: x = 0f(x) = x^2-4
Positive leading coefficient
Negative leading coefficientDownward-opening parabola with vertex at 0, 4, symmetry axis at horizontal coordinate 0, 2 real horizontal-axis intercepts at 2 and -2, and vertical-axis intercept 4.-6-5-4-3-2-1123456-6-5-4-3-2-112345xyvertex (0,4)root (2,0)root (-2,0)axis of symmetry: x = 0f(x) = -x^2+4
Negative leading coefficient

Minimum versus maximum

Positive leading coefficient

When \(a>0\), the arms rise and the vertex is the minimum. This does not mean every output is positive.

Negative leading coefficient

When \(a<0\), the arms fall and the vertex is the maximum. The sign describes direction, not the signs of every point.

Three useful forms

Quadratic forms and the information each reveals
FormStructureVisible immediatelyBest use
Standard\(ax^2+bx+c\)Vertical-axis intercept \((0,c)\)Coefficients and the symmetry formula
Vertex\(a(x-h)^2+k\)Vertex \((h,k)\)Maximum/minimum and transformations
Factored\(a(x-r_1)(x-r_2)\)Real roots \(r_1,r_2\)Horizontal-axis intercepts and Zero Product Property
Vertex from standard form
\[h=-\frac{b}{2a},\qquad k=f(h),\qquad \text{vertex}=(h,k)\]

Compute the horizontal coordinate first, then substitute it into the original function for the vertical coordinate.

Worked example

Find a vertex without guessing from a sketch

Find the vertex and symmetry axis of \(f(x)=3x^2-12x+7\).

  1. Identify coefficients

    Here \(a=3\), \(b=-12\), and \(c=7\).

  2. Find the axis

    \(h=-\frac{-12}{2(3)}=2\), so the axis is \(x=2\).

  3. Evaluate

    \(k=f(2)=3(2)^2-12(2)+7=-5\).

  4. Classify

    Because \(a>0\), the vertex is a minimum.

The vertex is \((2,-5)\), and the axis of symmetry is \(x=2\).

One quadratic viewed three ways

Equivalent forms of one original function

Standard

\(f(x)=2x^2-12x+10\), so the vertical-axis intercept is \(10\).

Vertex

\(f(x)=2(x-3)^2-8\), so the vertex is \((3,-8)\).

Factored

\(f(x)=2(x-1)(x-5)\), so the real roots are \(1\) and \(5\).

One quadratic, three formsUpward-opening parabola with vertex at 3, -8, symmetry axis at horizontal coordinate 3, 2 real horizontal-axis intercepts at 1 and 5, and vertical-axis intercept 10.-11234567-10-9-8-7-6-5-4-3-2-1123456789101112xyvertex (3,-8)root (1,0)root (5,0)vertical-axis intercept (0,10)axis of symmetry: x = 3f(x) = 2x^2-12x+10
One quadratic, three forms

Read a graph systematically

  1. Direction

    Decide whether the parabola opens upward or downward.

  2. Vertex

    Read the turning point and classify it as a minimum or maximum.

  3. Symmetry

    Use the vertex's horizontal coordinate for the vertical symmetry line.

  4. Intercepts

    Read horizontal-axis intercepts as real roots and the vertical-axis intercept as \(f(0)\).

  5. Consistency

    Check that symmetric points have equal output values.

Common mistakes and traps

  • In \(a(x-h)^2+k\), the vertex coordinate is \(h\), not the visible sign inside the parentheses.
  • The constant \(c\) is the vertical-axis intercept, not generally a root.
  • An upward-opening parabola can have negative output values near its vertex.
  • A parabola may have no real horizontal-axis intercepts.
Mini check

Connect factored form to symmetry

For \(g(x)=-2(x+1)(x-7)\), what is the symmetry axis?

  1. \(x=-4\)
  2. \(x=3\)
  3. \(x=4\)
  4. \(x=7\)
Show answer and explanation

Answer: \(x=3\)

The roots are \(-1\) and \(7\); their average is \(\frac{-1+7}{2}=3\).

Key takeaways

Key takeaways

What to remember

  • The sign of \(a\) controls opening direction.
  • Use \(h=-\frac{b}{2a}\) and \(k=f(h)\) for the vertex.
  • Standard, vertex, and factored forms reveal different graph features.
  • Real roots are exactly the horizontal-axis intercepts.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.