Transform a quadratic and find its key features

Base curve

\(\displaystyle y=x^{2}\)

A fountain design uses the displayed rule. What transformations turn the given base curve into the fountain stream?

Move h from -3 to 3 first and watch the vertex V and the readouts. Then flip s between -1 and 1. Watch the opening, the vertex height and the value one unit right of the vertex

What did h and s change, for every position of the sliders?

Quadratic transformations and features

Completed-square form shows the transformations of the base parabola and gives the vertex and axis of symmetry

1. 1Read completed-square form Match the displayed completed-square form to read the signed coefficient and both shifts \(\displaystyle y=-3(x+5)^{2}+6\Rightarrow a=-3,\ h=-5,\ k=6\)
2. 2Interpret the coefficient A negative a reflects the graph in the x-axis, and the size of a controls the width \(\displaystyle a=-3\Rightarrow\text{reflected, stretched by factor }3\)
3. 3Read the vertex and axis The vertex records the horizontal and vertical shifts; the symmetry axis is the vertical line through it \(\displaystyle V=(-5,6),\qquad x=-5\)
4. 4Find horizontal intercepts Set y to zero and solve for x, keeping both the positive and the negative square root \(\displaystyle y=-\frac{1}{2}(x-9)^2+18,\quad y=0\Rightarrow x=3\text{ or }x=15\)

Stream rule

\(\displaystyle y=-0.5(x-9)^{2}+18\)

Horizontal distance

\(\displaystyle x\text{ in metres}\)

Height

\(\displaystyle y\text{ in metres}\)

The given rule models a fountain stream. Horizontal distance and height are both measured in metres. What is the maximum point?

Read h and k from the completed-square form, and check whether the negative a gives a maximum or a minimum

Stream rule

\(\displaystyle y=-0.5(x-9)^{2}+18\)

Horizontal distance

\(\displaystyle x\text{ in metres}\)

Height

\(\displaystyle y\text{ in metres}\)

For the same fountain stream, find both horizontal positions where the water reaches ground level

Set the height to zero and solve, keeping both the positive and the negative square root

Rule

\(\displaystyle y=3(x+5)^{2}+6\)

Reversing the horizontal sign. One of these lines is right. Which one?

Turn features into a complete drawing

Work on paper or a blank drawing surface. Find the features, construct the whole labelled curve, and then review it. 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 quadratic graph

Use the completed-square rule to find every feature before drawing one smooth downward-opening parabola

1. 1Vertex and symmetry The squared term is zero at the horizontal shift; the negative coefficient makes this the maximum \(\displaystyle V=(1,3),\qquad x=1\)
2. 2Vertical intercept and a symmetric point Substitute zero and reflect the point across the symmetry axis \(\displaystyle f(0)=-2(0-1)^2+3=1,\qquad (0,1),\ (2,1)\)
3. 3Horizontal intercepts Set the output to zero and retain both square roots \(\displaystyle \begin{aligned}-2(x-1)^2+3&=0\\(x-1)^2&=\frac32\\x&=1\pm\sqrt{\frac32}\end{aligned}\)
4. 4Domain, range and ends Every real input is allowed, the maximum output is three, both ends fall and there are no asymptotes \(\displaystyle D=\mathbb R,\qquad R=(-\infty,3]\)

Rule

\(\displaystyle f(x)=-2(x-1)^2+3\)

Calculate the output at the vertical-axis intercept of the given worked parabola

Your complete hand-sketch task

On separate axes, draw the two complete graphs below. Label intercepts, domain and range, and the quadratic vertex and symmetry axis. Explain why the two displayed quadratic expressions give the same graph

1. 1Line Draw the complete straight line with its intercepts labelled \(\displaystyle y=3x-2\)
2. 2Quadratic Draw the whole parabola and compare its completed-square and expanded forms \(\displaystyle y=(x+2)^2-4,\qquad x^2+4x\)

Review both complete drawings

Compare your two drawings with these independently calculated features; both graphs continue across all real inputs

1. 1Line intercepts and gradient Substitute zero for each coordinate in turn; the coefficient of the input gives the gradient \(\displaystyle \begin{aligned}x=0&\Rightarrow y=-2\\y=0&\Rightarrow3x-2=0\\x&=\frac23,\qquad m=3\\(0,-2),\quad(\frac23,0)\end{aligned}\)
2. 2Line domain and range The non-horizontal line has no bound in either direction \(\displaystyle D=\mathbb R,\qquad R=\mathbb R\)
3. 3Quadratic vertex and axis Read the shifts from completed-square form; the positive coefficient gives a minimum \(\displaystyle y=(x+2)^2-4:\quad V=(-2,-4),\quad x=-2\)
4. 4Quadratic intercepts Set the output to zero and keep both square roots; the origin is also the vertical intercept \(\displaystyle \begin{aligned}(x+2)^2&=4\\x+2&=\pm2\\x&=-4\text{ or }0\\(-4,0),\quad(0,0)\end{aligned}\)
5. 5Quadratic domain and range Every real input is allowed and the squared term cannot be negative \(\displaystyle D=\mathbb R,\qquad R=[-4,\infty)\)
6. 6Equivalent expressions Expand the square and collect the constant terms \(\displaystyle (x+2)^2-4=x^2+4x+4-4=x^2+4x\)
7. 7Review 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