AI HL · 12.4
0 of 45 skills done
Graphs of polynomial equations
12.4.5
Screenshot and video to come
Screenshot and video to come
Move the slider for a. The curve goes up and down. The roots are where it meets the x-axis. Watch them appear, join and disappear.
Move the slider to move the curve.
Drag the orange point up and down to move the line y = k. The solutions of f(x) = k are the x-values where the line meets the curve.
Drag the point to move the line.
The graph of \(y = x^{4} + x^{3} - 10 x^{2} - 20 x - 8\) is shown. Solve \(x^{4} + x^{3} - 10 x^{2} - 20 x - 8 = 0\) and say which root is repeated.
Have a go first. Then open the working one line at a time.
Graph y = x^{4} + x^{3} - 10 x^{2} - 20 x - 8; read the x-intercepts
Roots are the x-intercepts of the graph.
From the calculator: x = −2.00, −0.562, 3.56
Three different x-values, although the equation has degree 4.
The graph touches the axis at x = −2 and turns back
A touch, not a cross, means a repeated root: (x + 2) is a factor twice.
g(−2) = 0 and g'(−2) = 0
A repeated root has the curve flat on the axis.
Solve \(x^{3} - 6 x = 3\) by drawing \(y = x^{3} - 6 x\) and \(y = 3\). Give the solutions to 3 significant figures.
Have a go first. Then open the working one line at a time.
Graph y = x^{3} - 6 x and y = 3
A solution of f(x) = 3 is an x-value where the curve is at height 3.
From the calculator: intersections at x = −2.15, −0.524, 2.67
The line cuts the curve three times, so there are three solutions.
Polynomial solver: x^{3} - 6 x - 3 = 0 gives the same three values
Moving 3 to the left side gives a cubic equal to zero. Same answers.
The graph of \(y = x^3 − 3x^2 + c\) moves up or down as the constant \(c\) changes. Find the positive value of \(c\) for which the graph touches the \(x\)-axis at its minimum point. For that \(c\), find all the roots.
Have a go first. Then open the working one line at a time.
f(x) = x³ − 3x², f′(x) = 3x² − 6x = 0, so x = 0 or x = 2
The maximum is at x = 0 and the minimum is at x = 2.
At x = 2: y = 8 − 12 + c = c − 4
The minimum touches the axis when this height is 0.
c − 4 = 0, so c = 4
The graph is lifted 4 units.
x³ − 3x² + 4 = (x - 2)**2*(x + 1) = 0, so x = −1 or x = 2 (repeated)
Check on the calculator: the graph crosses at −1 and touches at 2.
0 < c < 4: three roots. c = 0 or c = 4: two roots. c < 0 or c > 4: one root.
Between the minimum and the maximum heights, the axis cuts the curve three times.
Write \(y = x^2 + 10x + 25\) in the form \(y = (x − p)^2 + q\), sketch the graph and state the repeated root.
Have a go first. Then open the working one line at a time.
x² + 10x + 25 = (x + 5)² + 0
Half of 10 is 5, and 5² = 25 matches the constant exactly.
Vertex at (−5, 0)
q = 0, so the vertex is on the x-axis.
(x + 5)² = 0, so x = −5 (repeated)
The parabola touches the axis at one point and does not cross it.
The graph of \(y = x^4 − 5x^2 + 4\) is shown. It has four roots. What is the smallest number of real roots that a polynomial of degree 5 can have?
Have a go first. Then open the working one line at a time.
From the calculator: x = −2, −1, 1, 2
Four roots, the most that a degree 4 polynomial can have.
Degree 4: both ends go up. Degree 5: one end up, one end down
The sign of the highest power sets the ends.
Can have 0 real roots (the graph can sit above the axis)
For example, y = x⁴ + 1.
Always at least 1 real root, at most 5
The graph must cross the axis to get from one end to the other.