Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

Sketching y = a(x − h)² + k by Hand

2.3About 65 minutes

Learning goals

I can sketch y = a(x − h)² + k by applying transformations to y = x².
I can use the vertex and the step pattern multiplied by a to plot points quickly.
I can use symmetry to complete a sketch.
I can verify a sketch by substitution or with technology.

Words to know

sketch
A quick graph that shows the correct shape, vertex, direction and a few accurate points, without plotting a full table.

step pattern
From the vertex, over 1 and up a, then up 3a, then up 5a; reflected on the other side.

verify
To check that an answer is correct, for example by substituting a point or graphing with technology.

Sketching in under a minute

With vertex form you can sketch a parabola by hand, quickly and accurately, without a table of values. A sketch is not a perfect graph: it shows the correct vertex, direction, shape and a few accurate points, joined by a smooth curve. Engineers, scientists and students sketch first to know what to expect, then use technology to check.

The sketching method for y = a(x − h)² + k

Plot the vertex (h, k) and lightly draw the axis of symmetry x = h.
Use the step pattern multiplied by a. From the vertex, go 1 right and up a × 1; then 1 more right and up a × 3; then 1 more right and up a × 5. (If a is negative, "up" becomes "down".)
Reflect each point across the axis of symmetry to get the matching points on the left.
Join the points with a smooth curve and label the vertex and the equation.
Check one point by substituting it into the equation, or check the whole graph with technology.

Sketching y = −½(x − 3)² + 4 in three steps. Plot the vertex (3, 4) and the axis x = 3, use the steps −0.5, −1.5, −2.5 on each side, then draw a smooth curve.

Worked example

Example 1: Sketch y = −½(x − 3)² + 4

Sketch the graph by applying transformations to y = x², then verify with technology.

Transformations: reflection in the x-axis, vertical compression by a factor of ½, translation 3 right and 4 up.
Vertex (3, 4); axis x = 3; it opens down.
Steps: a × 1 = −0.5, a × 3 = −1.5, a × 5 = −2.5. From (3, 4): right 1 down 0.5 → (4, 3.5); right 1 down 1.5 → (5, 2); right 1 down 2.5 → (6, −0.5).
Reflect across x = 3: (2, 3.5), (1, 2), (0, −0.5). The point (0, −0.5) is the y-intercept. Check: −½(0 − 3)² + 4 = −4.5 + 4 = −0.5. ✓
Join the seven points with a smooth curve opening down.
Verify: type y = −0.5(x − 3)^2 + 4 into Desmos. The curve passes through (3, 4), (5, 2) and (0, −0.5).

A downward parabola with vertex (3, 4) through (2, 3.5), (4, 3.5), (1, 2), (5, 2), (0, −0.5) and (6, −0.5). It crosses the x-axis a little to the right of 0 and a little to the left of 6.

Sketching y = 3(x + 1)² − 2: from the vertex (−1, −2), go over 1 and up 3, then over 1 and up 9, and reflect across x = −1.

Worked example

Example 2: Sketch y = 3(x + 1)² − 2

Sketch the graph and state the y-intercept.

Vertical stretch by 3, translation 1 left and 2 down. Vertex (−1, −2), axis x = −1, opens up.
Steps × 3: up 3, then up 9 (then up 15, which goes off a small grid).
Right side: (0, −2 + 3) = (0, 1); (1, 1 + 9) = (1, 10).
Left side by symmetry: (−2, 1); (−3, 10).
The y-intercept is the point with x = 0: 1. Check: 3(0 + 1)² − 2 = 3 − 2 = 1. ✓

An upward, narrow parabola with vertex (−1, −2) through (−3, 10), (−2, 1), (0, 1) and (1, 10); y-intercept 1.

Worked example

Example 3: Sketching finds the zeros

Sketch y = (x − 2)² − 9 and use the sketch to find the zeros.

Vertex (2, −9); a = 1; steps up 1, 3, 5.
From (2, −9): (3, −8), (4, −5), (5, 0). Reflect: (1, −8), (0, −5), (−1, 0).
Two of the step points land exactly on the x-axis: (5, 0) and (−1, 0).
Check: (5 − 2)² − 9 = 9 − 9 = 0 ✓ and (−1 − 2)² − 9 = 9 − 9 = 0 ✓.

The zeros are −1 and 5. (When the zeros are not whole numbers, a sketch only estimates them; you will learn exact methods in Quadratic Expressions and Equations.)

Sketching tips

Always start from the vertex, never from the origin.
Make the steps from the vertex, not from the x-axis.
If a is a fraction, steps such as 0.5, 1.5, 2.5 are fine; or use bigger x-steps: for a = ½, going over 2 from the vertex gives ½ × 4 = 2.
Label the vertex and at least one other point with coordinates, so a reader can trust your sketch.

Just for fun

A sketch artist and a mathematician both draw parabolas. The difference? Only the mathematician says "up 1, up 3, up 5" out loud while doing it, and gets strange looks on the bus.

Hands-on activity

Sketch race with a check

You need: graph paper, a pencil, a timer, and a phone, tablet or computer with Desmos (free) for checking.

Set a timer for 2 minutes. Sketch as many of these as you can, using the vertex and the step pattern: y = (x − 1)² − 4, y = −2(x + 2)² + 3, y = 0.5(x − 4)², y = −(x + 1)² + 1.
After the timer, type each equation into Desmos and compare. For each sketch, award yourself 1 point for the correct vertex, 1 for the correct direction, and 1 for passing through the right step points.
Challenge a family member: give them a vertex-form equation and race to sketch it. Then have them describe it in words.

What to look for: y = (x − 1)² − 4: vertex (1, −4), through (0, −3), (2, −3), (−1, 0) and (3, 0). y = −2(x + 2)² + 3: vertex (−2, 3), through (−3, 1), (−1, 1), (−4, −5) and (0, −5). y = 0.5(x − 4)²: vertex (4, 0), through (3, 0.5), (5, 0.5), (2, 2) and (6, 2). y = −(x + 1)² + 1: vertex (−1, 1), through (0, 0) and (−2, 0).

Your turn

List the vertex and four more points for a sketch of y = (x + 3)² − 1.

List the vertex and four more points for a sketch of y = −2(x − 1)² + 5.

What are the first three steps from the vertex for y = 4(x − h)² + k?

Sketch y = 0.5x² − 2 and state its zeros.

For y = −(x − 4)² + 9, sketch and find the zeros.

Find the y-intercept of y = 2(x − 2)² − 3, then use symmetry to name another point at the same height.

More practice: warm-up

What is the first thing you plot when sketching y = a(x − h)² + k?

Steps for y = −x² from the vertex?

Steps for y = 2x²?

Vertex of y = 5(x − 2)² + 1?

Why do you reflect points across the axis of symmetry?

More practice: on your own

Sketch y = −3(x + 2)²: give the vertex, two points on each side and the number of zeros.

For y = ¼(x − 1)² + 2, use x-steps of 2 from the vertex to find points.

Sketch y = (x − 3)² − 1 and give its zeros, y-intercept and minimum value.

A student sketched y = 2(x − 1)² + 3 with points (2, 4) and (0, 4). What mistake did they make?

Without graphing, explain why the sketch of y = −0.5(x + 6)² − 2 never crosses the x-axis.

Challenge

Sketch y = −½(x − 3)² + 4 again. Estimate its zeros to one decimal place from your sketch, and check one with technology.

A parabola sketched from y = a(x − h)² + k passes through (1, 7), (3, −1) and (5, 7), and (3, −1) is the vertex. Find a, h and k.

Compare the sketches of y = 2(x − 1)² + 3 and y = 2(x − 1)² − 3. What single transformation maps the first sketch onto the second?

Check your understanding

Quiz

When sketching y = 2(x + 4)² − 1, which point do you plot first?

The first two steps from the vertex of y = −3(x − 1)² + 2 are…

Which point is on y = (x − 2)² + 1?

A sketch of y = −(x + 1)² + 4 has zeros at…

Which point is the mirror image of (5, 6) on a parabola with axis of symmetry x = 2?

Which is the best check of a hand sketch?

For parents and teachers

Big idea: Expectation 2.3 asks students to sketch y = a(x − h)² + k by hand by applying transformations to y = x² (the curriculum's sample is y = −½(x − 3)² + 4, verified with technology). The method is: vertex, step pattern times a, reflect, smooth curve, check.

How to help: Time a sketch race and check the results together in Desmos. Ask your teen to narrate: "Vertex at …, a is …, so the steps are …".

Common difficulties:

Starting the steps from the origin instead of the vertex.
Forgetting to multiply the steps by a, or forgetting that negative a means down.
Pointed vertices and straight arms. Encourage smooth, rounded curves.
Wrong sign of h when plotting the vertex.