Read key features off a graph, transform parent functions, build piecewise rules, and solve equations, inequalities, and systems — all checked on the HP Prime G2.
This chapter covers how a graph “talks” — its key features — and how to bend a parent
graph into a new one, stitch pieces together, and solve equations and systems. Each lesson
gives you a teach-the-tool Method and a one-line Quick-verify you can always run,
even when the full solve is off-limits on a test.
Problem types: Identify domain, range, x- and y-intercepts, increasing/decreasing intervals, and the average rate of change over an interval, by reading a graph.
🎣 Method (teach-the-tool)
Tool
Function app (Symbolic + Plot + Numeric views), with the Fcn analysis menu
Why this tool
Graphing the function lets the calculator confirm by hand-found features — where it crosses, where it turns, and where it rises or falls — instead of eyeballing a printed graph. The Numeric table reads domain/range values exactly, and Home computes the average rate of change so you don't slip an arithmetic sign.
Path
Apps ▸ Function ▸ Symb ▸ type F1(X) (use the X,θ,T,N key for X) ▸ Plot. In Plot view tap [Menu] ▸ [Fcn] ▸ Root (x-intercept) or Extremum (turning point). For the y-intercept press Num and read the X=0 row. Average rate of change: Home ▸ (F1(b)−F1(a))/(b−a) ▸ Enter.
📝 Worked example
Problem: Find the average rate of change of f(X)=X² over the interval [-2, 0]; then describe where f(X)=X²−4 increases and decreases.
Do this on the calculator:
AppsOpens the App Library — the screen full of app icons.
FunctionHighlight the Function app and press Enter to open it. It starts on the Symb (symbolic/equation) screen.
TypeF1(X)=X^2Type your function into the first slot, F1(X). Use the [X,θ,T,N] key for X and the x^y key for the power.
EnterSaves the definition for F1(X).
HomeSwitch to the Home screen for plain number-crunching.
Type(F1(0)−F1(-2))/(0−(-2))This is the average-rate-of-change formula (f(b)−f(a))/(b−a); use the (−) negative-sign key, not the minus key, for the negative numbers.
EnterThe screen prints -2 — the average rate of change, matching the hand answer (0−4)/(0−(-2)) = -2.
SymbGo back to the equation screen to change the function.
TypeF1(X)=X^2−4Edit F1(X) to the new function (tap [Edit] first if you need to change the existing text), then press Enter.
PlotDraws the graph — here a U-shaped parabola sitting 4 units down.
Menu▸Fcn▸ExtremumTap the [Menu] soft button, then [Fcn], then pick Extremum to find the turning point (the vertex).
You'll see: The Home screen prints -2 (the average rate of change). After Extremum, the status line reads Extremum: X=0, Y=-4 — the vertex (0, -4), so f decreases on (-∞, 0) and increases on (0, ∞).
✅ Quick-verify (check your hand-work)
PlotShow the graph of your function.
Menu▸Fcn▸RootTap [Menu], then [Fcn], then Root — it reads back the x-intercept you found by hand (move the tracer near the crossing first).
Menu▸Fcn▸ExtremumSame menu, pick Extremum — it reads back the turning point that splits where the graph rises from where it falls.
Problem types: Graph a transformed function from its parent — vertical/horizontal shifts f(x)+k and f(x−h), reflections −f(x) and f(−x), and stretches/compressions k·f(x); and write the equation of a transformed graph.
🎣 Method (teach-the-tool)
Tool
Function app (Symbolic + Plot views), overlaying two definitions
Why this tool
Define the parent in F1 and the transformed function in F2, then graph both at once. If your transformed equation is right, F2 sits exactly where the shift/stretch/reflection should put it relative to F1 — the overlay is the proof.
Path
Apps ▸ Function ▸ Symb ▸ F1(X)=parent ▸ Enter ▸ F2(X)=transformed ▸ Enter (keep both checked) ▸ Plot ▸ compare the two curves.
📝 Worked example
Problem: Graph the parent f(X)=|X| and the stretch g(X)=2·f(X)=2|X| together, and confirm the point (4, 8).
Do this on the calculator:
AppsOpens the App Library — the screen full of app icons.
FunctionHighlight the Function app and press Enter to open it. It starts on the Symb (equation) screen.
TypeF1(X)=ABS(X)Type the parent function into F1(X); ABS is the absolute-value command, so this makes a V shape.
EnterSaves F1(X).
TypeF2(X)=2*ABS(X)Type the stretched function into the second slot, F2(X). Leave both F1 and F2 check-marked so both will graph.
EnterSaves F2(X).
PlotDraws both curves — two V's sharing the point (0,0); F2 is the steeper one.
NumSwitch to the Numeric table that lists F1 and F2 values down the X column.
Type4With the cursor in the X column, type 4 to jump to the X=4 row; it reads F1=4 and F2=8, confirming the stretched point (4, 8).
You'll see: Plot shows two V's sharing the vertex (0,0); F2 is the steeper one. In the Num table the X=4 row reads F1=4 and F2=8, confirming the stretched point (4, 8).
✅ Quick-verify (check your hand-work)
PlotGraph the parent (F1) and your candidate (F2) together — they line up only where the transformation predicts; if F2 lands elsewhere, the equation is wrong.
NumSwitch to the Numeric table to read exact values.
Type4Type a point (e.g. 4) in the X column and check the table value against your hand answer.
Problem types: Graph a piecewise-defined function over its domain pieces, and write a piecewise rule from a graph (including step functions).
🎣 Method (teach-the-tool)
Tool
Function app with the PIECEWISE command (Plot + Numeric views)
Why this tool
Entering the rule as one PIECEWISE definition graphs every branch on its own interval, so the calculator draws the open/closed boundary behavior for you. Tracing or tabling the boundary x-values confirms each piece matches what you evaluated by hand.
Path
Apps ▸ Function ▸ Symb ▸ F1(X)=PIECEWISE(Case1, Test1, Case2, Test2, …) ▸ Enter ▸ Plot. (The brace template on the Templates-key palette enters the same thing visually.) Use Num, or trace in Plot, to read the value at each boundary x.
📝 Worked example
Problem: Graph f(X) = X+4 for -5 ≤ X < -2 and X²−2 for -2 ≤ X ≤ 5, and read the value at the boundary X=-2.
Do this on the calculator:
AppsOpens the App Library — the screen full of app icons.
FunctionHighlight the Function app and press Enter to open it. It starts on the Symb (equation) screen.
TypeF1(X)=PIECEWISE(X+4, X≥-5 AND X<-2, X²−2, X≥-2 AND X≤5)Type the whole rule into F1(X) using the PIECEWISE command — each piece is a pair: the formula, then the X-interval it applies on. (The brace template on the Templates key enters the same thing visually if you prefer.)
EnterSaves the piecewise definition.
PlotDraws a straight segment on [-5, -2) and a parabola arc on [-2, 5].
NumSwitch to the Numeric table to read exact values down the X column.
Type-2With the cursor in the X column type -2 (use the (−) negative-sign key) to jump to that row; F1 reads 2 — the second piece (-2)²−2=2, confirming the second piece is closed at the boundary.
You'll see: Plot draws a line segment on [-5, -2) and a parabola arc on [-2, 5]. In the Num table the X=-2 row reads F1=2 — the second piece (-2)²−2=2, confirming the second piece is closed at the boundary.
✅ Quick-verify (check your hand-work)
NumSwitch to the Numeric table.
Type-2Type each boundary x (e.g. -2) into the X column; the table value must equal the piece you hand-evaluated for that interval.
PlotTrace in Plot to confirm which side of the boundary is open (hollow dot) versus closed (filled dot).
Problem types: Solve a one-variable equation by graphing each side and finding where they meet; solve a one-variable inequality by graphing and reading off the region where one side is above/below the other.
🎣 Method (teach-the-tool)
Tool
Equations → Function app with Fcn ▸ Intersection. Inequalities → Advanced Graphing app (shaded region).
Why this tool
Set each side equal to y and graph both. For an equation, the x at the intersection is the solution — the Intersection tool pins it exactly instead of estimating. For an inequality, the Advanced Graphing app shades every (x,y) that makes the relation true, so you can read the solution interval and drop a test point in to confirm.
Path
Equation: Apps ▸ Function ▸ Symb ▸ F1(X)=left side ▸ F2(X)=right side ▸ Plot ▸ [Menu] ▸ [Fcn] ▸ Intersection ▸ pick the other curve. Inequality: Apps ▸ Advanced Graphing ▸ Symb ▸ V1: type the open sentence (relations palette = Shift+(=)) ▸ Plot ▸ read the shaded region; [Trace] ▸ Inside to walk inside it.
📝 Worked example
Problem: Solve |X−4| = ½X + 1 by graphing.
Do this on the calculator:
AppsOpens the App Library — the screen full of app icons.
FunctionHighlight the Function app and press Enter to open it. It starts on the Symb (equation) screen.
TypeF1(X)=ABS(X−4)Type the left side of the equation into F1(X); ABS is the absolute-value command.
EnterSaves F1(X).
TypeF2(X)=X/2+1Type the right side, ½X + 1, into F2(X). Leave both check-marked so both graph.
EnterSaves F2(X).
PlotDraws the V-shaped graph and the line — they cross twice.
Menu▸Fcn▸IntersectionTap [Menu], then [Fcn], then Intersection, and pick the other curve from the list. The status line reads Intersection: X=2, Y=2 — the first solution.
Menu▸Fcn▸IntersectionMove the tracer toward the second crossing with the arrow keys, then run Intersection again — it reads X=10, Y=6, the second solution. (Intersection returns only the crossing nearest the cursor, so you find each one separately.)
You'll see: The V-shaped graph and the line cross twice. The status line at the first crossing reads Intersection: X=2, Y=2; tracing right to the second gives Intersection: X=10, Y=6. The two solutions are X=2 and X=10.
✅ Quick-verify (check your hand-work)
HomeSwitch to the Home screen for plain number-crunching to test an inequality answer.
Type3^2−4Plug a test x from your answer region into the original inequality (e.g. X=3 for X²−4>0); use the x^y key for the power.
EnterThe screen prints 5 — since 5 > 0, x=3 really is in the solution region, so your region is right. (For an equation, just check the intersection x equals your algebra answer.)
Problem types: Solve a system of two (or three) linear equations; check a candidate solution; recognize no-solution (inconsistent) and infinite-solution cases.
🎣 Method (teach-the-tool)
Tool
Solve app (enter the equations) or CAS linsolve([...],[x,y]); graphical cross-check via Function app ▸ Fcn ▸ Intersection.
Why this tool
The Solve app and linsolve return the exact (x, y) — fast and reliable for many variables. Treat them as the answer-getter or the checker, then prove the result by substituting back: plug the solution into both equations in Home and confirm both sides are equal.
Path
Solve app: Apps ▸ Solve ▸ Symb ▸ E1: X+2*Y=3 ▸ E2: X−2*Y=4 (both checked) ▸ Num ▸ optionally seed a variable ▸ tap [Solve]. CAS: press CAS ▸ linsolve([X+2*Y=3, X−2*Y=4],[X,Y]) ▸ Enter. Graphical: Apps ▸ Function, solve each equation for Y into F1/F2, Plot ▸ [Menu] ▸ [Fcn] ▸ Intersection.
📝 Worked example
Problem: Solve the system x + 2y = 3 and x − 2y = 4.
Do this on the calculator:
CASPress the dedicated CAS key to open the CAS screen — the one that does exact, symbolic algebra (fractions stay as fractions).
Typelinsolve([X+2*Y=3, X−2*Y=4],[X,Y])Type the linsolve command: first the list of equations in square brackets, then the list of variables to solve for. (You can type the name with the Alpha keys, or pick it from the Toolbox ▸ CAS menu.)
EnterCAS returns [7/2, -1/4] — that is x = 7/2, y = -1/4.
You'll see: CAS returns [7/2, -1/4] — that is x = 7/2, y = -1/4. (Verify in Home: 7/2+2*(-1/4) ▸ Enter prints 3, and 7/2−2*(-1/4) ▸ Enter prints 4.)
✅ Quick-verify (check your hand-work)
HomeSwitch to the Home screen for plain number-crunching to substitute your answer back in.
Type7/2+2*(-1/4)Plug the solution into the first original equation; use the (−) negative-sign key for the negative number.
EnterPrints 3, which matches the right side of the first equation ✓.
Type7/2−2*(-1/4)Plug the same solution into the second original equation.
EnterPrints 4, matching the second equation ✓. Both true means the solution is confirmed.