Graphing quadratic inequalities worksheet - Free Printable

Let’s solve each problem step by step.

---

Problem 1:

We are given the inequality:
> y < 8x² - 5

We need to check which of the given points satisfies this inequality — meaning, when we plug in the x and y values, the statement should be TRUE.

Let’s test each option:

A) (1,4)
Plug x=1, y=4 into the inequality:
Is 4 < 8*(1)² - 5?
→ 4 < 8 - 5 → 4 < 3 → False

B) (0,10)
Plug x=0, y=10:
Is 10 < 8*(0)² - 5?
→ 10 < 0 - 5 → 10 < -5 → False

C) (0,-5)
Plug x=0, y=-5:
Is -5 < 8*(0)² - 5?
→ -5 < 0 - 5 → -5 < -5 → False (because it’s not *less than*, it’s equal — and the inequality is strict: “<”)

D) (1,2)
Plug x=1, y=2:
Is 2 < 8*(1)² - 5?
→ 2 < 8 - 5 → 2 < 3 → True

✔ So only D works.

---

Problem 2:

Inequality:
> y < 3x² - 7x + 3

This is a quadratic inequality. The graph of y = 3x² - 7x + 3 is a parabola opening upwards (since coefficient of x² is positive).

The inequality says y is less than the value on the parabola. That means we want all the points that lie below the parabola.

But the question asks: “the solution is ___ the parabola.”

Options:
A) on → no, because it’s strictly less than, not ≤
B) inside → for an upward-opening parabola, “inside” usually means above or between arms — but here we want below
C) outside → yes! For an upward-opening parabola, “outside” typically refers to the region below the curve (away from the vertex side)
D) both inside and outside → too vague

Actually, let’s think carefully: In standard terminology for parabolas:

- If you have y > ax²+bx+c (and a>0), the solution is “above” or “inside” the parabola (the U-shape contains the region).
- If you have y < ax²+bx+c (and a>0), the solution is “below” or “outside” the parabola.

So since our inequality is y < ... and parabola opens up, the solution region is outside the parabola.

✔ Correct answer: C) outside

---

Problem 3:

Inequality:
> y ≤ x² + x - 20

First, note: it’s “≤”, so the boundary (the parabola itself) is included → solid line.

Parabola: y = x² + x - 20

Find roots to sketch:
x² + x - 20 = 0
Factor: (x + 5)(x - 4) = 0 → roots at x = -5 and x = 4

Vertex is halfway between roots: x = (-5 + 4)/2 = -0.5
y-value at vertex: plug x = -0.5
y = (-0.5)² + (-0.5) - 20 = 0.25 - 0.5 - 20 = -20.25

So vertex around (-0.5, -20.25)

Since coefficient of x² is positive, parabola opens upward.

Inequality: y ≤ parabola → shade below the parabola (including the curve itself).

Now look at the two graphs:

Left graph: shaded everywhere EXCEPT between the arms → that would be y ≥ parabola (or something else) — actually, looking closely, it looks like the entire plane except the interior of the parabola is shaded — which might correspond to y ≥ ... if parabola opened down, but it doesn’t.

Wait — let me re-express:

Actually, left graph: the red area is everything except the narrow region between the two arms near the bottom — that suggests it's shading where y is very large or very small? No — wait, the parabola goes through (-5,0) and (4,0)? Let’s check:

At x = -5: y = (-5)^2 + (-5) - 20 = 25 -5 -20 = 0 → yes
At x = 4: y = 16 + 4 - 20 = 0 → yes

So parabola crosses x-axis at -5 and 4, vertex at (-0.5, -20.25)

If we shade y ≤ parabola, we shade BELOW the parabola — which includes the vertex area and extends downward.

Look at right graph: it shows a narrow shaded region between the two arms, going downward — that matches “below the parabola”.

Left graph: shaded almost everywhere except a small region near the bottom — that would be y ≥ parabola (if parabola opened up, shading above would be most of the plane).

But our inequality is y ≤ parabola → shade below → which is the narrow region between the arms, going down.

Also, since it’s “≤”, the line should be solid — both graphs show solid lines? Actually, in the image, both seem to have solid boundaries.

But the key is: which one shades the correct region?

Right graph: shaded region is bounded between the two arms, going down → that’s exactly where y ≤ x²+x-20 holds.

Left graph: shaded region is the rest of the plane — that would be y ≥ x²+x-20.

So correct graph is the right one.

✔ Answer: Right graph

---

Final Answers:

1) D
2) C
3) Right graph