from manim import * import numpy as np config.frame_width = 9 config.frame_height = 16 config.pixel_width = 1080 config.pixel_height = 1920 config.background_color = "#000000" CYAN = "#00E8FF" PINK = "#FF3DA8" GOLD = "#FFE14D" LIME = "#7CFF4D" AMBER = "#FFB020" LILAC = "#C5B4FF" AXIS = "#E2E8F0" MUTED = "#B7C3D4" SOFT = "#C5D4EA" X_MIN, X_MAX = -4.0, 3.2 Y_MIN, Y_MAX = -0.35, 4.55 Y_PLOT_LO = 0.028 def clipped_xrange(h): lo = h + np.log2(Y_PLOT_LO) hi = h + np.log2(Y_MAX * 0.97) a = max(X_MIN, float(lo)) b = min(X_MAX, float(hi)) if b - a < 0.45: b = min(X_MAX, a + 0.45) return [a, b] def make_badge(tex, color, sid_box, sid_text, font_size=26): label = MathTex(tex, font_size=font_size, color=color) box = SurroundingRectangle( label, buff=0.08, corner_radius=0.07, color=color, stroke_width=2.0, ) box.set_fill("#000000", 0.94) box label grp = VGroup(box, label) grp.set_z_index(8) return grp class TraslacionesHorizontalesExponencial(Scene): def construct(self): SAFE_X = config.frame_width / 2 - 0.35 SAFE_Y_BOTTOM = config.frame_height / 2 - 0.7 shared_reason = ( "The live exponential, both axes, the marked intercept and " "characteristic points, and the dashed y=0 asymptote share one " "set of axes so the horizontal shift can be read directly. " "Every 2^{x-h} has horizontal asymptote y=0, the x-axis itself." ) # ------------------------------------------------------------------ # Title # ------------------------------------------------------------------ t1 = Text("Traslaciones horizontales", font_size=32, weight=BOLD, color=WHITE) t2 = Text("en función exponencial creciente", font_size=25, color=SOFT) title = VGroup(t1, t2).arrange(DOWN, buff=0.24) title.to_edge(UP, buff=0.40) title.set_z_index(12) self.play(FadeIn(title, shift=DOWN * 0.15), run_time=1.1) self.wait(0.7) # ------------------------------------------------------------------ # Criterion band # ------------------------------------------------------------------ domain = MathTex( r"f:\mathbb{R}\rightarrow\mathbb{R}^{+}", font_size=32, color=SOFT, ) general = MathTex(r"f(x)=a^{x},\quad 1 SAFE_X - 0.08: asym_badge.shift(LEFT * (asym_badge.get_right()[0] - SAFE_X + 0.10)) if asym_badge.get_bottom()[1] < -SAFE_Y_BOTTOM: asym_badge.set_y(-SAFE_Y_BOTTOM + asym_badge.height / 2 + 0.04) def y_value(h): return 2 ** (-h) def make_yint_dot(): d = Dot(axes.c2p(0, y_value(h_tr.get_value())), radius=0.11, color=GOLD) d.set_z_index(6) return d def make_char_dot(): h = h_tr.get_value() op = 0.0 if abs(h) < 0.08 else min(1.0, (abs(h) - 0.08) / 0.18) d = Dot(axes.c2p(h, 1.0), radius=0.11, color=LIME) d.set_fill(LIME, opacity=op) d.set_stroke(width=0, opacity=op) d.set_z_index(6) return d yint_dot = always_redraw(make_yint_dot) char_dot = always_redraw(make_char_dot) seed_curve = plot_exp(0, color=CYAN, width=6.6) seed_yint = Dot(axes.c2p(0, 1), radius=0.11, color=GOLD) seed_yint.set_z_index(6) seed_yint # Left of the intercept: an increasing exponential occupies the # first quadrant there, so that pocket stays empty of the curve. badge_01 = make_badge( r"(0,1)", GOLD, "badge.01.box", "badge.01.text", font_size=26 ) badge_01.next_to(seed_yint, LEFT, buff=0.52) caption_anchor_y = axes.get_bottom()[1] - 0.95 def put_caption(mob): mob.set_x(0) mob.move_to([0, caption_anchor_y - mob.height / 2, 0]) if mob.get_bottom()[1] < -SAFE_Y_BOTTOM: mob.shift(UP * (-SAFE_Y_BOTTOM - mob.get_bottom()[1] + 0.04)) mob.set_z_index(11) return mob self.play( FadeIn(axes), FadeIn(xlab), FadeIn(ylab), FadeIn(x_ticks), FadeIn(y_ticks), run_time=1.15, ) self.play(Create(seed_curve), run_time=2.0) self.play(FadeIn(asym_line), FadeIn(asym_badge), run_time=0.9) self.play(GrowFromCenter(seed_yint), FadeIn(badge_01, scale=0.85), run_time=0.85) self.wait(0.55) no_x = Text( "No existe intersección con el eje x", font_size=24, color=MUTED, ) put_caption(no_x) self.play(FadeIn(no_x, shift=UP * 0.08), run_time=0.6) self.wait(1.8) self.play(FadeOut(no_x), run_time=0.45) family = MathTex(r"f_{h}(x)=2^{x-h}", font_size=42, color=WHITE) put_caption(family) self.play(FadeIn(family, shift=UP * 0.08), run_time=0.75) self.wait(1.6) self.play(FadeOut(family), run_time=0.4) self.add(curve, yint_dot, char_dot) self.remove(seed_curve, seed_yint) # ------------------------------------------------------------------ # Helpers for each horizontal shift # ------------------------------------------------------------------ def formula_tex(expr): m = MathTex(expr, font_size=46, color=WHITE) m.next_to(live_domain, DOWN, buff=0.32) m.set_x(0) m.set_z_index(12) return m def yint_spec(h): if abs(h) < 1e-6: return r"(0,1)" if abs(h - 1) < 1e-6: return r"(0,\tfrac{1}{2})" if abs(h - 2) < 1e-6: return r"(0,\tfrac{1}{4})" if abs(h + 1) < 1e-6: return r"(0,2)" if abs(h + 2) < 1e-6: return r"(0,4)" return r"(0,1)" def char_spec(h): if abs(h) < 1e-6: return None if abs(h - 1) < 1e-6: return r"(1,1)" if abs(h - 2) < 1e-6: return r"(2,1)" if abs(h + 1) < 1e-6: return r"(-1,1)" if abs(h + 2) < 1e-6: return r"(-2,1)" return None def build_yint_badge(h, tag): tex = yint_spec(h) b = make_badge(tex, GOLD, f"badge.yint.{tag}.box", f"badge.yint.{tag}.text") anchor = Dot(axes.c2p(0, y_value(h)), radius=0.11) b.next_to(anchor, LEFT, buff=0.52) if h >= 1.5: b.shift(UP * 0.20) return b def build_char_badge(h, tag): tex = char_spec(h) if tex is None: return None b = make_badge(tex, LIME, f"badge.char.{tag}.box", f"badge.char.{tag}.text") anchor = Dot(axes.c2p(h, 1.0), radius=0.11) b.next_to(anchor, UP, buff=0.44) return b def dir_group(h_from, h_to, chip_msg): going_right = h_to > h_from end = RIGHT * 1.15 if going_right else LEFT * 1.15 arr = Arrow( ORIGIN, end, buff=0, color=AMBER, stroke_width=6.0, max_tip_length_to_length_ratio=0.28, max_stroke_width_to_length_ratio=10, ) arr chip = Text(chip_msg, font_size=24, color=AMBER) grp = VGroup(arr, chip).arrange(DOWN, buff=0.18) put_caption(grp) return grp def f0_tex(h): mapping = { 0: r"f(0)=2^{0}=1", 1: r"f(0)=2^{-1}=\tfrac{1}{2}", 2: r"f(0)=2^{-2}=\tfrac{1}{4}", -1: r"f(0)=2^{1}=2", -2: r"f(0)=2^{2}=4", } key = int(round(h)) return put_caption(MathTex(mapping[key], font_size=32, color=GOLD)) current_formula = formula current_ybadge = badge_01 current_cbadge = None tag_i = 0 def play_shift(h_to, new_expr, chip_msg, runtime=3.3, show_asym_note=False): nonlocal current_formula, current_ybadge, current_cbadge, tag_i tag_i += 1 h_from = h_tr.get_value() chip = dir_group(h_from, h_to, chip_msg) nxt_formula = formula_tex(new_expr) fade_out = [FadeIn(chip, shift=UP * 0.06)] if current_ybadge is not None: fade_out.append(FadeOut(current_ybadge)) if current_cbadge is not None: fade_out.append(FadeOut(current_cbadge)) self.play(*fade_out, run_time=0.45) self.play( h_tr.animate.set_value(h_to), ReplacementTransform(current_formula, nxt_formula), run_time=runtime, rate_func=smooth, ) current_formula = nxt_formula new_y = build_yint_badge(h_to, tag_i) new_c = build_char_badge(h_to, tag_i) appear = [FadeIn(new_y, scale=0.9)] if new_c is not None: appear.append(FadeIn(new_c, scale=0.9)) self.play(*appear, run_time=0.55) current_ybadge = new_y current_cbadge = new_c self.play(FadeOut(chip), run_time=0.30) comp = f0_tex(h_to) self.play(FadeIn(comp, shift=UP * 0.06), run_time=0.45) self.wait(1.35) self.play(FadeOut(comp), run_time=0.35) if show_asym_note: note = Text( "La asíntota horizontal no cambia", font_size=24, color=PINK, ) put_caption(note) self.play(FadeIn(note, shift=UP * 0.06), run_time=0.5) self.wait(1.7) self.play(FadeOut(note), run_time=0.4) self.wait(0.35) # ------------------------------------------------------------------ # Shifts to the right, return, then left # ------------------------------------------------------------------ play_shift( 1.0, r"f(x)=2^{x-1}", "h = 1 → 1 unidad a la derecha", runtime=3.4, show_asym_note=True, ) play_shift( 2.0, r"f(x)=2^{x-2}", "h = 2 → 2 unidades a la derecha", runtime=3.2, ) play_shift( 0.0, r"f(x)=2^{x}", "regreso a la función original", runtime=2.6, ) ident = MathTex(r"2^{x+1}=2^{x-(-1)}", font_size=36, color=AMBER) put_caption(ident) self.play(FadeIn(ident, shift=UP * 0.06), run_time=0.65) self.wait(1.8) self.play(FadeOut(ident), run_time=0.35) play_shift( -1.0, r"f(x)=2^{x+1}", "h = -1 → 1 unidad a la izquierda", runtime=3.3, ) play_shift( -2.0, r"f(x)=2^{x+2}", "h = -2 → 2 unidades a la izquierda", runtime=3.2, ) # ------------------------------------------------------------------ # Conclusion — fade the live criterion out first, then bring the # boxed general form in, so the two never occupy the same band. # ------------------------------------------------------------------ fade_end = [FadeOut(current_formula), FadeOut(live_domain)] if current_ybadge is not None: fade_end.append(FadeOut(current_ybadge)) if current_cbadge is not None: fade_end.append(FadeOut(current_cbadge)) self.play(*fade_end, run_time=0.55) boxed_form = MathTex(r"f(x)=a^{x-h},\quad 10", font_size=32, color=AMBER), Text("→ h unidades a la derecha", font_size=24, color=WHITE), ).arrange(RIGHT, buff=0.32) row_l = VGroup( MathTex(r"h<0", font_size=32, color=AMBER), Text("→ |h| unidades a la izquierda", font_size=24, color=WHITE), ).arrange(RIGHT, buff=0.32) rules = VGroup(row_r, row_l).arrange(DOWN, buff=0.22) put_caption(rules) if rules.get_bottom()[1] < -SAFE_Y_BOTTOM: rules.shift(UP * (-SAFE_Y_BOTTOM - rules.get_bottom()[1] + 0.06)) self.play(FadeIn(rules, shift=UP * 0.08), run_time=0.7) self.wait(0.4) self.play(h_tr.animate.set_value(1.6), run_time=2.4, rate_func=smooth) self.play(h_tr.animate.set_value(-1.6), run_time=3.4, rate_func=smooth) self.play(h_tr.animate.set_value(0.0), run_time=2.2, rate_func=smooth) self.wait(0.35) self.play(FadeOut(rules), run_time=0.4) asym_line_txt = MathTex( r"\text{Asíntota horizontal: }y=0", font_size=32, color=PINK, ) asym_box = SurroundingRectangle( asym_line_txt, buff=0.12, corner_radius=0.09, color=PINK, stroke_width=2.2, ) asym_box.set_fill("#000000", 0.55) asym_stay = Text("permanece constante", font_size=24, color=PINK) asym_concl = VGroup(VGroup(asym_box, asym_line_txt), asym_stay).arrange( DOWN, buff=0.20 ) asym_box asym_line_txt put_caption(asym_concl) self.play(FadeIn(asym_concl, shift=UP * 0.08), run_time=0.7) self.wait(1.8) self.play(FadeOut(asym_concl), run_time=0.4) final_a = Text("El cambio dentro del exponente", font_size=26, color=WHITE) final_b = Text("produce una traslación horizontal", font_size=26, color=CYAN) final = VGroup(final_a, final_b).arrange(DOWN, buff=0.26) put_caption(final) self.play(FadeIn(final, shift=UP * 0.08), run_time=0.75) self.wait(2.4) self.play(FadeOut(final), run_time=0.5) self.wait(0.6)

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