C3.2; E3.2 - Zeros, vertices and intersections

C3.2; E3.2 - Zeros, vertices and intersections

A graph can reveal where a function is zero, where it turns and where it meets another graph. Learn to locate these features with a graphic display calculator, then check that its coordinates answer the question. These are the shared C3.2 and E3.2 graph skills.

Zeros

Where does f(x)=x2−2x−8f(x)=x^2-2x-8 meet the horizontal axis, and how low does its curve go? A well-chosen window makes these features visible; the calculator's graph tools locate them more accurately than a cursor read-off.

Zero of a function

A zero is an input for which the output is zero: f(x)=0f(x)=0. A zero rr gives an intercept (r,0)(r,0) on the xx-axis. The zero is a number; the intercept is a coordinate pair.

Enter the complete function with brackets where needed. Set a window showing the relevant crossings. Select the zero/root feature from the graph menu and isolate the chosen crossing: calculators may ask for lower and upper bounds, or a nearby cursor position. Repeat at every crossing. A table near each answer provides a separate check. A curve may also touch the axis at a zero without crossing it.

[DIAGRAM: asset_name: Quadratic zeros and minimum; asset_slug: quadratic_features; caption: The two zeros are inputs; the vertex has two coordinates.; description: Exact graph of y equals x squared minus two x minus eight over minus four to six. Its intercepts are labelled minus two zero and four zero, and its minimum is labelled one minus nine. Axes have numbered scales.]
The two zeros are inputs

Worked example 1: two zeros

Question. Use a GDC to find the zeros of f(x)=x2−2x−8f(x)=x^2-2x-8.

Plan. Show both crossings, then isolate each with the zero feature.

  1. Enter Y1=X2−2X−8Y_1=X^2-2X-8. Use −4≤x≤6-4\le x\le6, −12≤y≤12-12\le y\le12.
  2. Locate the left crossing between −3-3 and −1-1: the zero feature returns x=−2x=-2.
  3. Repeat between 3 and 5: it returns x=4x=4.

Answer. The zeros are −2-2 and 4.

Check. f(−2)=4+4−8=0f(-2)=4+4-8=0 and f(4)=16−8−8=0f(4)=16-8-8=0. Both visible crossings have been located.

The next curve has no integer zeros. Keep the calculator's full values until the final rounding.

Worked example 2: approximate zeros

Question. Find the zeros of g(x)=x2−2x−5g(x)=x^2-2x-5, correct to 3 significant figures.

Plan. Locate both crossings, keeping extra digits for checking.

  1. Enter Y1=X2−2X−5Y_1=X^2-2X-5, using −3≤x≤5-3\le x\le5, −8≤y≤10-8\le y\le10.
  2. Isolate the crossings in (−2,−1)(-2,-1) and (3,4)(3,4).
  3. The graph tool gives approximately −1.449489743-1.449489743 and 3.4494897433.449489743.

Answer. x=−1.45x=-1.45 or x=3.45x=3.45, to 3 significant figures.

Check. Substitution of the unrounded values gives outputs close to zero. The rounded values each give 0.00250.0025, a small rounding residual rather than an exact zero.

The requested accuracy applies to the answer, not the intermediate calculator value.

Watch out

A zero is not the yy-intercept. The yy-intercept uses x=0x=0; finding a zero means finding an input that makes y=0y=0.

Use the graph feature for your next read-off.

Extrema and vertices

A curve's turning point has both an input and an output: its horizontal position and its height.

Local maximum and local minimum

At a local maximum, the function's value is at least as high as nearby values. At a local minimum, it is at least as low as nearby values. A quadratic's turning point is its vertex: a minimum for an upward-opening curve and a maximum for a downward-opening curve.

Use the graph's minimum or maximum feature. Isolate the turning point with bounds or a cursor neighbourhood as your calculator requires. Read both coordinates, and compare outputs on each side. The top or bottom edge of the screen is a window boundary, not automatically a turning point. An unfamiliar curve can have more than one local extremum, so inspect the whole requested interval.

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