Connect a factorised higher-degree rule and its graph

What should be read from each linear factor first?

Read a factorised polynomial

A factorised polynomial displays its roots and multiplicities directly. Use the degree and leading coefficient for the outer ends, then substitute zero to find the vertical intercept

1. 1Read a root from its factor Set each linear factor equal to zero to find its horizontal-axis intercept \(\displaystyle \left(x-4\right)=0\Rightarrow x=4\)
2. 2Keep each multiplicity The exponent on a factor is the multiplicity of its root \(\displaystyle \left(x+2\right)^3\Rightarrow x=-2\text{ has multiplicity }3\)
3. 3Use degree and leading sign Degree parity determines whether the ends agree, and the leading sign determines their direction \(\displaystyle \text{degree }4\text{ and leading coefficient }-2\Rightarrow\text{both ends fall}\)
4. 4Multiply factor values at zero The vertical intercept is found by substituting zero into every factor and multiplying the values \(\displaystyle P(0)=2\times(0-3)^2\times(0+1)=18\)

Evaluate the repeated factor when x=0

Find the value of P(0)

Which line correctly evaluates the expression?

Turn features into a complete drawing

The complete drawings and explanations are ungraded; the checked entry assesses one feature

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics without calculus Substitute labelled points, use factors and signs between roots, compare symmetry and both ends, and use tables or simple inequalities to check the shape. Distinguish a local extremum from a bound on the whole domain. A plotted model is feedback, not your submitted drawing

A complete quartic from squares and symmetry

For f(x)=(x²−1)², zeros at −1 and 1 are double roots. The curve touches the axis at the global minima (−1,0) and (1,0). It is even, so reflect the left and right halves in the y-axis. Between −1 and 1, 0≤x²≤1 implies 0≤(x²−1)²≤1, with the local maximum (0,1). Outside those roots the squared magnitude grows as |x| grows, so both tails rise. Domain R and range [0,∞); the local maximum is not a global maximum. Draw a smooth W, not a pair of disconnected parabolas. These facts follow from squares and inequalities, without derivatives

Rule

\(\displaystyle f(x)=(x^2-1)^2\)

Find the y-intercept of the worked quartic

Your complete hand-sketch task

Draw g(x)=(x²−4)² on R. Label roots and touching behaviour, all local extrema, symmetry, domain/range and both ends. Justify the extrema using squares and the interval −2≤x≤2. Then draw h(x)=1−(x²−4)² separately by reflecting g in the x-axis and shifting up 1. Find all four exact roots of h, its extrema and its range; decide whether each root crosses or touches. Verify a non-root point on each graph

Review both complete drawings

For g, double roots/global minima are (−2,0) and (2,0), the local maximum/y-intercept is (0,16), symmetry is the y-axis, domain R and range [0,∞). Both tails rise; g(1)=9. On [−2,2], 0≤x²≤4 implies 0≤(x²−4)²≤16. For h, the global maxima are (−2,1),(2,1), the local minimum/y-intercept is (0,−15), symmetry is the y-axis, domain R and range (−∞,1]. Its four simple roots are −√5, −√3, √3, √5, each a crossing. It is positive only on (−√5,−√3) and (√3,√5), and negative on the remaining intervals. Both tails fall; h(1)=−8. The local minimum −15 is not a global minimum

1. 1Find the roots of h Set h equal to zero, then solve the two possible square values \(\displaystyle 1-(x^2-4)^2=0\Rightarrow(x^2-4)^2=1\Rightarrow x^2=3\text{ or }x^2=5\Rightarrow x=\pm\sqrt3\text{ or }x=\pm\sqrt5\)
2. 2Review the whole curve Compare every labelled feature, branch and end direction with your own drawing. Annotate corrections and discuss discrepancies with a teacher or peer