Graphs of Quadratic Equations Practice

Read the sign of the squared-term coefficient before doing any coordinate calculations.

The leading coefficient is \(a=2\). Its sign shows the parabola opens upward, so its vertex is a minimum.

Read the sign of the squared-term coefficient before doing any coordinate calculations.

The leading coefficient is \(a=-3\). Its sign shows the parabola opens downward, so its vertex is a maximum.

Read the sign of the squared-term coefficient before doing any coordinate calculations.

The leading coefficient is \(a=1\). Its sign shows the parabola opens upward, so its vertex is a minimum.

Read the sign of the squared-term coefficient before doing any coordinate calculations.

The leading coefficient is \(a=-1\). Its sign shows the parabola opens downward, so its vertex is a maximum.

Read the sign of the squared-term coefficient before doing any coordinate calculations.

The leading coefficient is \(a=5\). Its sign shows the parabola opens upward, so its vertex is a minimum.

Substitute the signed value of the linear coefficient into the negative numerator.

Use \(x=-\frac{b}{2a}\): \(x=-\frac{-6}{2(1)}=3\).

Substitute the signed value of the linear coefficient into the negative numerator.

Use \(x=-\frac{b}{2a}\): \(x=-\frac{8}{2(2)}=-2\).

Substitute the signed value of the linear coefficient into the negative numerator.

Use \(x=-\frac{b}{2a}\): \(x=-\frac{12}{2(-3)}=2\).

Substitute the signed value of the linear coefficient into the negative numerator.

Use \(x=-\frac{b}{2a}\): \(x=-\frac{-16}{2(4)}=2\).

Substitute the signed value of the linear coefficient into the negative numerator.

Use \(x=-\frac{b}{2a}\): \(x=-\frac{-4}{2(-2)}=-1\).

Find the horizontal coordinate first, then substitute it into the original function.

The symmetry coordinate is \(x=-\frac{b}{2a}=2\). Evaluating the function gives \(h(2)=-3\), so the vertex is \((2,-3)\).

Find the horizontal coordinate first, then substitute it into the original function.

The symmetry coordinate is \(x=-\frac{b}{2a}=-2\). Evaluating the function gives \(h(-2)=-5\), so the vertex is \((-2,-5)\).

Find the horizontal coordinate first, then substitute it into the original function.

The symmetry coordinate is \(x=-\frac{b}{2a}=3\). Evaluating the function gives \(h(3)=7\), so the vertex is \((3,7)\).

Find the horizontal coordinate first, then substitute it into the original function.

The symmetry coordinate is \(x=-\frac{b}{2a}=2\). Evaluating the function gives \(h(2)=-4\), so the vertex is \((2,-4)\).

Find the horizontal coordinate first, then substitute it into the original function.

The symmetry coordinate is \(x=-\frac{b}{2a}=-2\). Evaluating the function gives \(h(-2)=13\), so the vertex is \((-2,13)\).

Match the expression to vertex form instead of reading the inside sign literally.

Vertex form is \(a(x-h)^2+k\), so the vertex is \((h,k)=(3,-5)\). The sign inside the parentheses is opposite the coordinate's sign.

Match the expression to vertex form instead of reading the inside sign literally.

Vertex form is \(a(x-h)^2+k\), so the vertex is \((h,k)=(-4,2)\). The sign inside the parentheses is opposite the coordinate's sign.

Match the expression to vertex form instead of reading the inside sign literally.

Vertex form is \(a(x-h)^2+k\), so the vertex is \((h,k)=(-2,-7)\). The sign inside the parentheses is opposite the coordinate's sign.

Match the expression to vertex form instead of reading the inside sign literally.

Vertex form is \(a(x-h)^2+k\), so the vertex is \((h,k)=(5,1)\). The sign inside the parentheses is opposite the coordinate's sign.

Match the expression to vertex form instead of reading the inside sign literally.

Vertex form is \(a(x-h)^2+k\), so the vertex is \((h,k)=(6,-9)\). The sign inside the parentheses is opposite the coordinate's sign.

In a factor written as a difference, the root has the opposite visible sign.

Set each factor equal to zero. The roots are \(-3\) and \(5\), so the intercepts are \(( -3,0 )\) and \(( 5,0 )\).

In a factor written as a difference, the root has the opposite visible sign.

Set each factor equal to zero. The roots are \(2\) and \(7\), so the intercepts are \(( 2,0 )\) and \(( 7,0 )\).

In a factor written as a difference, the root has the opposite visible sign.

Set each factor equal to zero. The roots are \(-4\) and \(-1\), so the intercepts are \(( -4,0 )\) and \(( -1,0 )\).

In a factor written as a difference, the root has the opposite visible sign.

Set each factor equal to zero. The roots are \(0\) and \(6\), so the intercepts are \(( 0,0 )\) and \(( 6,0 )\).

In a factor written as a difference, the root has the opposite visible sign.

Set each factor equal to zero. The roots are \(-5\) and \(3\), so the intercepts are \(( -5,0 )\) and \(( 3,0 )\).

In standard form, the constant is the output when the input is zero.

At the vertical axis, \(x=0\). Therefore \(f(0)=11\), and the intercept is \((0,11)\).

In standard form, the constant is the output when the input is zero.

At the vertical axis, \(x=0\). Therefore \(f(0)=-5\), and the intercept is \((0,-5)\).

In standard form, the constant is the output when the input is zero.

At the vertical axis, \(x=0\). Therefore \(f(0)=4\), and the intercept is \((0,4)\).

In standard form, the constant is the output when the input is zero.

At the vertical axis, \(x=0\). Therefore \(f(0)=-8\), and the intercept is \((0,-8)\).

In standard form, the constant is the output when the input is zero.

At the vertical axis, \(x=0\). Therefore \(f(0)=6\), and the intercept is \((0,6)\).

Average the roots; do not subtract them or use their sum without dividing by two.

Real roots are symmetric about the vertex. Their average is \(\frac{-5+3}{2}=-1\).

Average the roots; do not subtract them or use their sum without dividing by two.

Real roots are symmetric about the vertex. Their average is \(\frac{1+9}{2}=5\).

Average the roots; do not subtract them or use their sum without dividing by two.

Real roots are symmetric about the vertex. Their average is \(\frac{-8+-2}{2}=-5\).

Average the roots; do not subtract them or use their sum without dividing by two.

Real roots are symmetric about the vertex. Their average is \(\frac{-1+7}{2}=3\).

Average the roots; do not subtract them or use their sum without dividing by two.

Real roots are symmetric about the vertex. Their average is \(\frac{4+12}{2}=8\).

Expand the binomial square before distributing the leading coefficient.

Expand the square and distribute \(1\): the result is \(f(x)=x^2-4x-5\). Coefficient comparison confirms the forms match.

Expand the binomial square before distributing the leading coefficient.

Expand the square and distribute \(2\): the result is \(f(x)=2x^2+4x+5\). Coefficient comparison confirms the forms match.

Expand the binomial square before distributing the leading coefficient.

Expand the square and distribute \(-1\): the result is \(f(x)=-x^2+8x-10\). Coefficient comparison confirms the forms match.

Expand the binomial square before distributing the leading coefficient.

Expand the square and distribute \(3\): the result is \(f(x)=3x^2+12x+11\). Coefficient comparison confirms the forms match.

Expand the binomial square before distributing the leading coefficient.

Expand the square and distribute \(-2\): the result is \(f(x)=-2x^2-12x-13\). Coefficient comparison confirms the forms match.

Read the turning point and count horizontal-axis intersections; do not infer root count from opening direction alone.

The exact vertex is \((1,-4)\). Because the leading coefficient is positive, its vertical coordinate is the minimum. Solving \(f(x)=0\) gives 2 real roots, matching the plotted intercepts.

Read the turning point and count horizontal-axis intersections; do not infer root count from opening direction alone.

The exact vertex is \((2,9)\). Because the leading coefficient is negative, its vertical coordinate is the maximum. Solving \(f(x)=0\) gives 2 real roots, matching the plotted intercepts.

Read the turning point and count horizontal-axis intersections; do not infer root count from opening direction alone.

The exact vertex is \((0,-8)\). Because the leading coefficient is positive, its vertical coordinate is the minimum. Solving \(f(x)=0\) gives 2 real roots, matching the plotted intercepts.

Read the turning point and count horizontal-axis intersections; do not infer root count from opening direction alone.

The exact vertex is \((-3,0)\). Because the leading coefficient is positive, its vertical coordinate is the minimum. Solving \(f(x)=0\) gives 1 real root, matching the plotted intercepts.

Read the turning point and count horizontal-axis intersections; do not infer root count from opening direction alone.

The exact vertex is \((2,4)\). Because the leading coefficient is positive, its vertical coordinate is the minimum. Solving \(f(x)=0\) gives 0 real roots, matching the plotted intercepts.

For a point-value question, evaluate the function algebraically rather than estimating pixels.

Direct substitution gives \(f(0)=2(0)^2+-8(0)+6=6\). The graph is generated from the same equation, so \(( 0,6 )\) lies on it.

For a point-value question, evaluate the function algebraically rather than estimating pixels.

Direct substitution gives \(f(1)=-2(1)^2+-4(1)+6=0\). The graph is generated from the same equation, so \(( 1,0 )\) lies on it.

For a point-value question, evaluate the function algebraically rather than estimating pixels.

Direct substitution gives \(f(2)=3(2)^2+6(2)+-9=15\). The graph is generated from the same equation, so \(( 2,15 )\) lies on it.

For a point-value question, evaluate the function algebraically rather than estimating pixels.

Direct substitution gives \(f(-2)=-1(-2)^2+2(-2)+8=0\). The graph is generated from the same equation, so \(( -2,0 )\) lies on it.

For a point-value question, evaluate the function algebraically rather than estimating pixels.

Direct substitution gives \(f(5)=1(5)^2+-10(5)+21=-4\). The graph is generated from the same equation, so \(( 5,-4 )\) lies on it.