17.6 Quadratic Curve When It Opens Upwards

🟦 Concept

A quadratic equation can be written in the form:$$y=ax^2+bx+c$$

The value of a tells us whether the quadratic curve opens upwards or downwards.

When:$$a>0$$

the curve opens upwards and has a U-shape ∪.

An upward-opening quadratic has a minimum turning point, which is the lowest point on the curve.

Remember:
Positive aa → Opens upwards ∪ → Minimum

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🟩 Example Math

Consider the quadratic equation:$$y=x^2-4x+3$$

Compare it with:$$y=ax^2+bx+c$$

Therefore:$$a=1,\quad b=-4,\quad c=3$$

Since:$$a=1>0$$

the curve opens upwards.

Before plotting the curve, let’s calculate its important features.

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1. Find the Turning Point

The x-coordinate of the turning point is found using:$$x=-\frac{b}{2a}$$

Substitute $a=1$ and $b=-4$:$$x=-\frac{-4}{2(1)}$$$$x=2$$

Now substitute $x=2$ into the original equation:$$y=(2)^2-4(2)+3$$$$y=4-8+3$$$$y=-1$$

So the turning point is:$$(2,-1)$$

Because the curve opens upwards, this is a minimum turning point.

The axis of symmetry is:$$x=2$$

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2. Find the Roots

The roots occur where the curve meets the x-axis.

At the x-axis:$$y=0$$

Therefore:$$x^2-4x+3=0$$

Factorise:$$(x-1)(x-3)=0$$

So:$$x=1 \quad \text{or} \quad x=3$$

The roots are:$$(1,0) \quad \text{and} \quad (3,0)$$

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3. Find the y-intercept

The y-intercept occurs where the curve meets the y-axis.

At the y-axis:$$x=0$$

Substitute $x=0$:$$y=(0)^2-4(0)+3$$$$y=3$$

So the y-intercept is:$$(0,3)$$

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🟨 Now Plot the Quadratic Curve

We now have the important information needed to plot the graph:

Feature	Result
Equation	$y=x^2-4x+3$
Value of $a$	$1$ — positive
Direction	Opens upwards ∪
Turning point	$(2,-1)$
Type	Minimum
Axis of symmetry	$x=2$
Roots	$(1,0)$ and $(3,0)$
y-intercept	$(0,3)$

Plot these points and join them with a smooth U-shaped curve.

Because the graph is symmetrical about:$$x=2$$

the two sides of the parabola should have the same shape.

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🟪 Range

The lowest point on the graph is the turning point:$$(2,-1)$$

Therefore, the smallest possible value of $y$ is:$$-1$$

The curve then continues upwards.

So the range is:$$y\geq-1$$

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💡 Exam Tip

For:$$y=ax^2+bx+c$$

look at the sign of aa first.

a>0a>0 → Opens upwards ∪ → Minimum

a<0a<0 → Opens downwards ∩ → Maximum

This is one of the quickest ways to understand the shape of a quadratic graph before you even draw it.

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⚠️ Common Mistake

Don’t confuse the turning point with the minimum value.

For this example:

Turning point: $(2,-1)$

Minimum value: $-1$

The turning point is a coordinate, while the minimum value is the lowest y-value.

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⭐ Quick Summary

For:$$y=x^2-4x+3$$

we have:

a=1>0a=1>0 → curve opens upwards ∪

Turning point: $(2,-1)$

Axis of symmetry: $x=2$

Roots: $(1,0)$ and $(3,0)$

y-intercept: $(0,3)$

Range: $y\geq-1$

Remember

Positive aa → Upwards ∪ → Minimum

now