A mathematics problem uses Parabola vertex form. Given coefficient (a) = 0.6000; x-value (x) = 0.0000; vertex x (h) = -3.0000; vertex y (k) = 1.0000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3

   $$List the givens: coefficient (a) = 0.6000, x-value (x) = 0.0000, vertex x (h) = -3.0000, vertex y (k) = 1.0000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = 6.4000$$

6. Step 6 — Check: returning y = 6.4000 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 12.8000 — kept a factor of two that cancels in the correct rearrangement.
• 3.2000 — dropped that same factor in the other direction.
• 7.0400 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Parabola

A vertical curve profile is modeled as a parabola. Given curvature coefficient (a) = 0.2000 1/m; station offset (x) = 28.5000 m; vertex station (h) = 1.1000 m; vertex elevation (k) = 8.8000 m, determine the elevation (y) in m.

Start with the thinking

• The governing relation printed in this handbook section is Parabola (vertex form).
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A vertical curve profile is modeled as a parabola.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a(x - h)^2 + k$$

2. Step 2 — Rearrange symbolically for y:

   $$y = y = a(x - h)^2 + k$$

3. Step 3 — List the givens: curvature coefficient (a) = 0.2000 1/m, station offset (x) = 28.5000 m, vertex station (h) = 1.1000 m, vertex elevation (k) = 8.8000 m.

4. Step 4 — Substitute the given values:

   $$y = y = 0.2000(28.5000 - 1.1000)^2 + 8.8000$$

5. Step 5 — Evaluate:

   $$y = 159.0\ \text{m}$$

6. Step 6 — Check: returning y = 159.0 m to

   $$y = a(x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 317.9 — kept a factor of two that cancels in the correct rearrangement.
• 79.4760 — dropped that same factor in the other direction.
• 174.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Parabola)

A mathematics problem uses Parabola vertex form. Given x-value (x) = -3.0000; vertex x (h) = -2.5000; vertex y (k) = 2.5000; y-value (y) = 11.2000, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that a stands alone on the left-hand side.

3. Step 3

   $$List the givens: x-value (x) = -3.0000, vertex x (h) = -2.5000, vertex y (k) = 2.5000, y-value (y) = 11.2000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$a = 34.8000$$

6. Step 6 — Check: returning a = 34.8000 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 69.6000 — kept a factor of two that cancels in the correct rearrangement.
• 17.4000 — dropped that same factor in the other direction.
• 38.2800 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Parabola

A vertical curve profile is modeled as a parabola. Given elevation (y) = 12.5000 m; station offset (x) = 27.5000 m; vertex station (h) = 6.1000 m; vertex elevation (k) = 9.9000 m, determine the curvature coefficient (a) in 1/m.

Start with the thinking

• The governing relation printed in this handbook section is Parabola (vertex form).
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A vertical curve profile is modeled as a parabola.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a(x - h)^2 + k$$

2. Step 2 — Rearrange symbolically for a:

   $$a = a = \dfrac{y - k}{(x - h)^2}$$

3. Step 3 — List the givens: elevation (y) = 12.5000 m, station offset (x) = 27.5000 m, vertex station (h) = 6.1000 m, vertex elevation (k) = 9.9000 m.

4. Step 4 — Substitute the given values:

   $$a = a = \dfrac{12.5000 - 9.9000}{(27.5000 - 6.1000)^2}$$

5. Step 5 — Evaluate:

   $$a = 0.0057\ \text{1/m}$$

6. Step 6 — Check: returning a = 0.0057 1/m to

   $$y = a(x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0114 — kept a factor of two that cancels in the correct rearrangement.
• 0.0028 — dropped that same factor in the other direction.
• 0.0062 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Parabola)

A mathematics problem uses Parabola vertex form. Given coefficient (a) = 2.0000; x-value (x) = 2.5000; vertex x (h) = -1.5000; y-value (y) = -38.8000, determine the vertex y (k).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that k stands alone on the left-hand side.

3. Step 3 — List the givens: coefficient (a) = 2.0000, x-value (x) = 2.5000, vertex x (h) = -1.5000, y-value (y) = -38.8000.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$k = -70.8000$$

6. Step 6 — Check: returning k = -70.8000 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -141.6 — kept a factor of two that cancels in the correct rearrangement.
• -35.4000 — dropped that same factor in the other direction.
• -77.8800 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Parabola

A vertical curve profile is modeled as a parabola. Given elevation (y) = 12.8000 m; curvature coefficient (a) = 0.3000 1/m; station offset (x) = 8.4000 m; vertex station (h) = 8.2000 m, determine the vertex elevation (k) in m.

Start with the thinking

• The governing relation printed in this handbook section is Parabola (vertex form).
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A vertical curve profile is modeled as a parabola.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a(x - h)^2 + k$$

2. Step 2 — Rearrange symbolically for k:

   $$k = k = y - a(x - h)^2$$

3. Step 3 — List the givens: elevation (y) = 12.8000 m, curvature coefficient (a) = 0.3000 1/m, station offset (x) = 8.4000 m, vertex station (h) = 8.2000 m.

4. Step 4 — Substitute the given values:

   $$k = k = 12.8000 - 0.3000(8.4000 - 8.2000)^2$$

5. Step 5 — Evaluate:

   $$k = 12.7880\ \text{m}$$

6. Step 6 — Check: returning k = 12.7880 m to

   $$y = a(x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 25.5760 — kept a factor of two that cancels in the correct rearrangement.
• 6.3940 — dropped that same factor in the other direction.
• 14.0668 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Parabola)

A mathematics problem uses Parabola vertex form. Given coefficient (a) = 2.8000; x-value (x) = 4.0000; vertex x (h) = 1.5000; vertex y (k) = -1.0000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3

   $$List the givens: coefficient (a) = 2.8000, x-value (x) = 4.0000, vertex x (h) = 1.5000, vertex y (k) = -1.0000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = 16.5000$$

6. Step 6 — Check: returning y = 16.5000 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 33.0000 — kept a factor of two that cancels in the correct rearrangement.
• 8.2500 — dropped that same factor in the other direction.
• 18.1500 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Parabola

A vertical curve profile is modeled as a parabola. Given curvature coefficient (a) = 0.2000 1/m; station offset (x) = 17.5000 m; vertex station (h) = 1.3000 m; vertex elevation (k) = 12.3000 m, determine the elevation (y) in m.

Start with the thinking

• The governing relation printed in this handbook section is Parabola (vertex form).
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A vertical curve profile is modeled as a parabola.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a(x - h)^2 + k$$

2. Step 2 — Rearrange symbolically for y:

   $$y = y = a(x - h)^2 + k$$

3. Step 3 — List the givens: curvature coefficient (a) = 0.2000 1/m, station offset (x) = 17.5000 m, vertex station (h) = 1.3000 m, vertex elevation (k) = 12.3000 m.

4. Step 4 — Substitute the given values:

   $$y = y = 0.2000(17.5000 - 1.3000)^2 + 12.3000$$

5. Step 5 — Evaluate:

   $$y = 64.7880\ \text{m}$$

6. Step 6 — Check: returning y = 64.7880 m to

   $$y = a(x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 129.6 — kept a factor of two that cancels in the correct rearrangement.
• 32.3940 — dropped that same factor in the other direction.
• 71.2668 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Parabola)

A mathematics problem uses Parabola vertex form. Given x-value (x) = 3.0000; vertex x (h) = 2.5000; vertex y (k) = 3.0000; y-value (y) = -17.1000, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that a stands alone on the left-hand side.

3. Step 3

   $$List the givens: x-value (x) = 3.0000, vertex x (h) = 2.5000, vertex y (k) = 3.0000, y-value (y) = -17.1000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$a = -80.4000$$

6. Step 6 — Check: returning a = -80.4000 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -160.8 — kept a factor of two that cancels in the correct rearrangement.
• -40.2000 — dropped that same factor in the other direction.
• -88.4400 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Parabola

A vertical curve profile is modeled as a parabola. Given elevation (y) = 32.8000 m; station offset (x) = 14.0000 m; vertex station (h) = 8.0000 m; vertex elevation (k) = 8.9000 m, determine the curvature coefficient (a) in 1/m.

Start with the thinking

• The governing relation printed in this handbook section is Parabola (vertex form).
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• A vertical curve profile is modeled as a parabola.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a(x - h)^2 + k$$

2. Step 2 — Rearrange symbolically for a:

   $$a = a = \dfrac{y - k}{(x - h)^2}$$

3. Step 3 — List the givens: elevation (y) = 32.8000 m, station offset (x) = 14.0000 m, vertex station (h) = 8.0000 m, vertex elevation (k) = 8.9000 m.

4. Step 4 — Substitute the given values:

   $$a = a = \dfrac{32.8000 - 8.9000}{(14.0000 - 8.0000)^2}$$

5. Step 5 — Evaluate:

   $$a = 0.6639\ \text{1/m}$$

6. Step 6 — Check: returning a = 0.6639 1/m to

   $$y = a(x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.3278 — kept a factor of two that cancels in the correct rearrangement.
• 0.3319 — dropped that same factor in the other direction.
• 0.7303 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Parabola)